Rama

(i) The Given expression is a² + 9a + 20.
By comparing the given expression a² + 9a + 20 with the basic expression x^2 + ax + b.
Here, a = 1, b = 9, and c = 20.
The sum of two numbers is m + n = b = 9 = 5 + 4.
The product of two number is m * n = a * c = 1 * (20) = 20 = 5 * 4
From the above two instructions, we can write the values of two numbers m and n as 5 and 4.
Then, a² + 9a + 20 = a² +5a + 4a + 20.
= a (a+ 5) + 4(a + 5).
Factor out the common terms.
(a + 5) (a + 4)

Then, a² + 9a + 20 = (a + 5) (a + 4).

(ii) The Given expression is m² + 15m + 54.
By comparing the given expression m² + 15m + 54 with the basic expression x^2 + ax + b.
Here, a = 1, b = 15, and c = 54.
The sum of two numbers is m + n = b = 15 = 6 + 9.
The product of two number is m * n = a * c = 1 * (54) = 54 = 6 * 9
From the above two instructions, we can write the values of two numbers m and n as 6 and 9.
Then, m² + 15m + 54 = m² + 6m + 9m + 54.
= m (m + 6) + 9(m + 6).
Factor out the common terms.
(m + 6) (m + 9)

Then, m² + 15m + 54 = (m + 6) (m + 9).

(iii) The Given expression is y² + 3y – 4.
By comparing the given expression y² + 3y – 4 with the basic expression x^2 + ax + b.
Here, a = 1, b = 3, and c = -4.
The sum of two numbers is m + n = b = 3 = 4 – 1.
The product of two number is m * n = a * c = 1 * (-4) = -4 = 1 * -4
From the above two instructions, we can write the values of two numbers m and n as 1 and -4.
Then, y² + 3y – 4 = y² + 4y -y – 4.
= y (y + 4) – 1(y + 4).
Factor out the common terms.
(y + 4) (y – 1)

Then, y² + 3y – 4 = (y + 4) (y – 1).

(iv) The Given expression is n² + 2n – 24.
By comparing the given expression n² + 2n – 24 with the basic expression x^2 + ax + b.
Here, a = 1, b = 2, and c = -24.
The sum of two numbers is m + n = b = 2 = 6 – 4.
The product of two number is m * n = a * c = 1 * (-24) = -24 = 6 * -4
From the above two instructions, we can write the values of two numbers m and n as 6 and -4.
Then, n² + 2n – 24 = n² + 6n – 4n – 24.
= n (n + 6) – 4(n + 6).
Factor out the common terms.
(n + 6) (n – 4)

Then, n² + 2n – 24 = (n + 6) (n – 4).

(v) The Given expression is x² – 5x + 4.
By comparing the given expression x² – 5x + 4 with the basic expression x^2 + ax + b.
Here, a = 1, b = -5, and c = 4.
The sum of two numbers is m + n = b = -5 = -1 – 4.
The product of two number is m * n = a * c = 1 * (4) = 4 = -1 * -4
From the above two instructions, we can write the values of two numbers m and n as -1 and -4.
Then, x² – 5x + 4 = x² – x – 4x + 4.
= x (x – 1) – 4(x – 1).
Factor out the common terms.
(x – 1) (x – 4)

Then, x² – 5x + 4 = (x – 1) (x – 4).

(vi) The Given expression is a² – 15a + 14.
By comparing the given expression a² – 15a + 14 with the basic expression x^2 + ax + b.
Here, a = 1, b = -15, and c = 14.
The sum of two numbers is m + n = b = -15 = -1 – 14.
The product of two number is m * n = a * c = 1 * (14) = 14 = -1 * -14
From the above two instructions, we can write the values of two numbers m and n as -1 and -14.
Then, a² – 15a + 14 = a² – a – 14a + 14.
= a (a – 1) – 14(a – 1).
Factor out the common terms.
(a – 1) (a – 14)

Then, a² – 15a + 14 = (a – 1) (a – 14).

(i) The Given expression is a² + 3a – 10.
By comparing the given expression a² + 3a – 10 with the basic expression x^2 + ax + b.
Here, a = 1, b = 3, and c = -10.
The sum of two numbers is m + n = b = 3 = 5 – 2.
The product of two number is m * n = a * c = 1 * (-10) = -10 = 5 * -2
From the above two instructions, we can write the values of two numbers m and n as 5 and -2.
Then, a² + 3a – 10 = a² + 5a – 2a – 10.
= a (a + 5) – 2(a + 5).
Factor out the common terms.
(a + 5) (a – 2)

Then, a² + 3a – 10 = (a + 5) (a – 2).

(ii) The Given expression is m² – 18m – 63.
By comparing the given expression m² – 18m – 63 with the basic expression x^2 + ax + b.
Here, a = 1, b = -18, and c = -63.
The sum of two numbers is m + n = b = -18 = 3 – 21.
The product of two number is m * n = a * c = 1 * (-63) = -63 = 3 * -21
From the above two instructions, we can write the values of two numbers m and n as 3 and -21.
Then, m² – 18m – 63 = m² + 3m – 21m – 63.
= m (m + 3) – 21(m + 3).
Factor out the common terms.
(m + 3) (m – 21)

Then, m² – 18m – 63 = (m + 3) (m – 21).

(iii) The Given expression is a² + 6a + 8.
By comparing the given expression a² + 6a + 8 with the basic expression x^2 + ax + b.
Here, a = 1, b = 6, and c = 8.
The sum of two numbers is m + n = b = 6 = 2 + 4.
The product of two number is m * n = a * c = 1 * (8) = 8 = 2 * 4
From the above two instructions, we can write the values of two numbers m and n as 2 and 4.
Then, a² + 6a + 8 = a² + 2a + 4a + 8.
= a (a + 2) + 4(a + 2).
Factor out the common terms.
(a + 2) (a + 4)

Then, a² + 6a + 8 = (a + 2) (a + 4).

(iv) The Given expression is a² + 12a + 32.
By comparing the given expression a² + 12a + 32 with the basic expression x^2 + ax + b.
Here, a = 1, b = 12, and c = 32.
The sum of two numbers is m + n = b = 12 = 8 + 4.
The product of two number is m * n = a * c = 1 * (32) = 32 = 8 * 4
From the above two instructions, we can write the values of two numbers m and n as 8 and 4.
Then, a² + 12a + 32 = a² + 8a + 4a + 32.
= a (a + 8) + 4(a + 8).
Factor out the common terms.
(a + 8) (a + 4)

Then, a² + 12a + 32 = (a + 8) (a + 4).

(v) The Given expression is x² – 8x + 15.
By comparing the given expression x² – 8x + 15 with the basic expression x^2 + ax + b.
Here, a = 1, b = -8, and c = 15.
The sum of two numbers is m + n = b = -8 = – 3 – 5.
The product of two number is m * n = a * c = 1 * (15) = 15 = -3 * -5
From the above two instructions, we can write the values of two numbers m and n as -3 and -5.
Then, x² – 8x + 15 = x² – 3x – 5x + 15.
= x (x – 3) – 5(x – 3).
Factor out the common terms.
(x – 3) (x – 5)

Then, x² – 8x + 15 = (x – 3) (x – 5).

(vi) The Given expression is m² – 12m + 35.
By comparing the given expression m² – 12m + 35 with the basic expression x^2 + ax + b.
Here, a = 1, b = -12, and c = 35.
The sum of two numbers is m + n = b = -12 = – 5 – 7.
The product of two number is m * n = a * c = 1 * (35) = 35 = -5 * -7
From the above two instructions, we can write the values of two numbers m and n as -5 and -7.
Then, m² – 12m + 35 = m² – 5m – 7m + 35.
= m (m – 5) – 7(m – 5).
Factor out the common terms.
(m – 7) (m – 5)

Then, m² – 12m + 35 = (m – 7) (m – 5).

(i) The Given expression is m² – 4m – 12.
By comparing the given expression m² – 4m – 12 with the basic expression x^2 + ax + b.
Here, a = 1, b = -4, and c = -12.
The sum of two numbers is m + n = b = -4 = 2 – 6.
The product of two number is m * n = a * c = 1 * (-12) = -12 = 2 * -6
From the above two instructions, we can write the values of two numbers m and n as 2 and -6.
Then, m² – 4m – 12 = m² – 2m + 6m – 12.
= m (m – 2) + 6(m – 2).
Factor out the common terms.
(m – 2) (m + 6)

Then, m² – 4m – 12 = (m – 2) (m + 6).

(ii) The Given expression is a² – 4a – 45.
By comparing the given expression a² – 4a – 45 with the basic expression x^2 + ax + b.
Here, a = 1, b = -4, and c = -45.
The sum of two numbers is m + n = b = -4 = 5 – 9.
The product of two number is m * n = a * c = 1 * (-45) = -45 = 5 * -9
From the above two instructions, we can write the values of two numbers m and n as 5 and -9.
Then, a² – 4a – 45 = a² + 5a – 9a – 45.
= a (a + 5) – 9(a + 5).
Factor out the common terms.
(a + 5) (a – 9)

Then, a² – 4a – 45 = (a + 5) (a – 9).

(iii) The Given expression is x² + 15x + 56.
By comparing the given expression x² + 15x + 56 with the basic expression x^2 + ax + b.
Here, a = 1, b = 15, and c = 56.
The sum of two numbers is m + n = b = 15 = 8 + 7.
The product of two number is m * n = a * c = 1 * (56) = 56 = 8 * 7
From the above two instructions, we can write the values of two numbers m and n as 8 and 7.
Then, x² + 15x + 56 = x² + 8x + 7x + 56.
= x (x + 8) + 7(x + 8).
Factor out the common terms.
(x + 8) (x + 7)

Then, x² + 15x + 56 = (x + 8) (x + 7).

(iv) The Given expression is p² – 13p + 36.
By comparing the given expression p² – 13p + 36 with the basic expression x^2 + ax + b.
Here, a = 1, b = -13, and c = 36.
The sum of two numbers is m + n = b = -13 = -9 – 4.
The product of two number is m * n = a * c = 1 * (36) = 36 = -9 * -4
From the above two instructions, we can write the values of two numbers m and n as -9 and -4.
Then, p² – 13p + 36 = p² – 9p – 4p + 36.
= p (p – 9) – 4(p – 9).
Factor out the common terms.
(p – 9) (p – 4)

Then, p² – 13p + 36 = (p – 9) (p – 4).

(v) The Given expression is q² + 5q – 24.
By comparing the given expression q² + 5q – 24 with the basic expression x^2 + ax + b.
Here, a = 1, b = 5, and c = -24.
The sum of two numbers is m + n = b = 5 = -3 + 8.
The product of two number is m * n = a * c = 1 * (-24) = -24 = -3 * 8
From the above two instructions, we can write the values of two numbers m and n as -3 and 8.
Then, q² + 5q – 24 = q² – 3q + 8q – 24.
= q (q – 3) + 8(q – 3).
Factor out the common terms.
(q – 3) (q + 8)

Then, q² + 5q – 24 = (q – 3) (q + 8).

(vi) The Given expression is r² + 17r – 84.
By comparing the given expression r² + 17r – 84 with the basic expression x^2 + ax + b.
Here, a = 1, b = 17, and c = -84.
The sum of two numbers is m + n = b = 17 = 21 – 4.
The product of two number is m * n = a * c = 1 * (-84) = -84 = 21 * -4
From the above two instructions, we can write the values of two numbers m and n as 21 and -4.
Then, r² + 17r – 84 = r² + 21r -4r – 84.
= r (r + 21) – 4(r + 21).
Factor out the common terms.
(r + 21) (r – 4)

Then, r² + 17r – 84 = (r + 21) (r – 4).

(vii) The Given expression is a² – 15a + 44.
By comparing the given expression a² – 15a + 44 with the basic expression x^2 + ax + b.
Here, a = 1, b = -15, and c = 44.
The sum of two numbers is m + n = b = -15 = -11 – 4.
The product of two number is m * n = a * c = 1 * (44) = 44 = -11 * -4
From the above two instructions, we can write the values of two numbers m and n as -11 and -4.
Then, a² – 15a + 44 = a² – 11a – 4a + 44.
= a (a – 11) – 4(a – 11).
Factor out the common terms.
(a – 11) (a – 4)

Then, a² – 15a + 44 = (a – 11) (a – 4).

(viii) The Given expression is m² – 5m – 24.
By comparing the given expression m² – 5m – 24 with the basic expression x^2 + ax + b.
Here, a = 1, b = -5, and c = -24.
The sum of two numbers is m + n = b = -5 = 3 – 8.
The product of two number is m * n = a * c = 1 * (-24) = -24 = 3 * -8
From the above two instructions, we can write the values of two numbers m and n as 3 and -8.
Then, m² – 5m – 24 = m² + 3m – 8m – 24.
= m (m + 3) – 8(m – 3).
Factor out the common terms.
(m – 3) (m – 8)

Then, m² – 5m – 24 = (m – 3) (m – 8).

(ix) The Given expression is x² – 4x – 77.
By comparing the given expression x² – 4x – 77 with the basic expression x^2 + ax + b.
Here, a = 1, b = -4, and c = -77.
The sum of two numbers is m + n = b = -4 = -11 + 7.
The product of two number is m * n = a * c = 1 * (-77) = -77 = -11 * 7
From the above two instructions, we can write the values of two numbers m and n as -11 and 7.
Then, x² – 4x – 77 = x² – 11x +7x – 77.
= x (x – 11) + 7(x – 11).
Factor out the common terms.
(x – 11) (x + 7)

Then, x² – 4x – 77 = (x – 11) (x + 7).

(x) The Given expression is a² – 12a + 20.
By comparing the given expression a² – 12a + 20 with the basic expression x^2 + ax + b.
Here, a = 1, b = -12, and c = 20.
The sum of two numbers is m + n = b = -12 = -10 – 2.
The product of two number is m * n = a * c = 1 * (20) = 20 = -10 * -2
From the above two instructions, we can write the values of two numbers m and n as -10 and -2.
Then, a² – 12a + 20 = a² – 10a – 2a + 20.
= a (a – 10) – 2(a – 10).
Factor out the common terms.
(a – 10) (a – 2)

Then, a² – 12a + 20 = (a – 10) (a – 2).

1. Given expression is a² – 36
Rewrite the given expression in the form of a² – b².
(a)² – (6)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a and b = 6
[a + 6] [a – 6]

The final answer is [a + 6] [a – 6]

2. Given expression is 4x² – 9
Rewrite the given expression in the form of a² – b².
(2x)² – (3)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 2x and b = 3
[2x + 3] [2x – 3]

The final answer is [2x + 3] [2x – 3]

3. Given expression is 81 – 49a²
Rewrite the given expression in the form of a² – b².
(9)² – (7a)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 9 and b = 7a
[9 + 7a] [9 – 7a]

The final answer is [9 + 7a] [9 – 7a]

4. Given expression is 4a² – 9b²
Rewrite the given expression in the form of a² – b².
(2a)² – (3b)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 2a and b = 3b
[2a + 3b] [2a – 3b]

The final answer is [2a + 3b] [2a – 3b]

5. Given expression is 16m² – 225n²
Rewrite the given expression in the form of a² – b².
(4m)² – (15n)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 4m and b = 15n
[4m + 15n] [4m – 15n]

The final answer is [4m + 15n] [4m – 15n]

6. Given expression is 9a²b² – 25
Rewrite the given expression in the form of a² – b².
(3ab)² – (5)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 3ab and b = 5
[3ab + 5] [3ab – 5]

The final answer is [3ab + 5] [3ab – 5]

7. Given expression is 16a² – ¹/₁₄₄
Rewrite the given expression in the form of a² – b².
(4a)² – (1/12)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 4a and b = 1/12
[4a + 1/12] [4a – 1/12]

The final answer is [4a + 1/12] [4a – 1/12]

8. Given expression is (2x + 3y)² – 16z²
Rewrite the given expression in the form of a² – b².
(2x + 3y)² – (4z)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 2x + 3y and b = 4z
[2x + 3y + 4z] [2x + 3y – 4z]

The final answer is [2x + 3y + 4z] [2x + 3y – 4z]

9. Given expression is 1 – (m – n)²
Rewrite the given expression in the form of a² – b².
(1)² – (m – n)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 1 and b = m – n
[1 + m – n] [1 – (m – n)]
[1 + m – n] [1 – m + n]

The final answer is [1 + m – n] [1 – m + n]

10. Given expression is 9(a + b)² – a²
Rewrite the given expression in the form of a² – b².
(3(a + b))² – (a)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 3(a + b) and b = a
[3a + 3b + a] [3a + 3b – a]
[4a + 3b] [2a + 3b]

The final answer is [4a + 3b] [2a + 3b]

11. Given expression is 25(x + y)² – 16(x – y)²
Rewrite the given expression in the form of a² – b².
(5(x + y))² – (4(x – y))²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 5(x + y) and b = 4(x – y)
[5x + 5y + 4x – 4y] [5x + 5y – 4x + 4y]
[9x + y] [x + 9y]

The final answer is [9x + y] [x + 9y]

12. Given expression is 20x² – 45y²
Rewrite the given expression in the form of a² – b².
5{(2x)² – (3y)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 2x and b = 3y
5{[2x + 3y] [2x – 3y]}

The final answer is 5{[2x + 3y] [2x – 3y]}

13. Given expression is a³ – 64a
Rewrite the given expression in the form of a² – b².
a{(a)² – (8)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a and b = 8
a{[a + 8] [a – 8]}

The final answer is a{[a + 8] [a – 8]}

14. Given expression is 12a² – 27
Rewrite the given expression in the form of a² – b².
3{(2a)² – (3)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 2a and b = 3
3{[2a + 3] [2a – 3]}

The final answer is 3{[2a + 3] [2a – 3]}

15. Given expression is 3a⁵ – 48a³
Rewrite the given expression in the form of a² – b².
3a³{(a)² – (4)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a and b = 4
3a³{[a + 4] [a – 4]}

The final answer is 3a³{[a + 4] [a – 4]}

16. Given expression is 63x²y² – 7
Rewrite the given expression in the form of a² – b².
7{(3xy)² – (1)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 3xy and b = 1
7{[3xy + 1] [3xy – 1]}

The final answer is 7{[3xy + 1] [3xy – 1]}

17. Given expression is m² – 2mn + n² – r²
Rewrite the given expression in the form of a² – b².
m² – 2mn + n² is in the form of a² – 2ab + b²
{(m – n)² – (r)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = m – n and b = r
{[m – n + r] [m – n – r]}

The final answer is 7{[3xy + 1] [3xy – 1]}

18. Given expression is a² – b² – 2ab – 1
Rewrite the given expression in the form of a² – b².
a² – b² – 2ab is in the form of a² – 2ab + b²
{(a – b)² – (1)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a – b and b = 1
{[a – b + 1] [a – b – 1]}

The final answer is {[a – b + 1] [a – b – 1]}

19. Given expression is 9a² – b² + 4b – 4
9a² – b² + 4b – 4 = 9a² – (b² – 4b + 2²)
Rewrite the given expression in the form of a² – b².
b² – 4b + 2² is in the form of a² – 2ab + b²
{(3a)² – (b – 2)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 3a and b = b – 2
{[3a + b – 2] [3a – b + 2]}

The final answer is {[3a + b – 2] [3a – b + 2]}

1. Given expression is a² – 2ab + b² – c²
Rewrite the given expression in the form of a² – b².
a² – b² – 2ab is in the form of a² – 2ab + b²
{(a – b)² – (c)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a – b and b = c
{[a – b + c] [a – b – c]}

The final answer is {[a – b + c] [a – b – c]}

2. Given expression is 25 – x² – y² -2xy
Rewrite the given expression in the form of a² – b².
25 – x² – y² -2xy = 25 – (x² + y² + 2xy)
x² + y² + 2xy is in the form of a² + 2ab + b²
{(5)² – (x + y)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 5 and b = x + y
{[5 + x + y] [5 – x – y]}

The final answer is {[5 + x + y] [5 – x – y]}

3. Given expression is 16b³ – 4b
Rewrite the given expression in the form of a² – b².
b{(4b)² – (2)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 4b and b = 2
b{[4b + 2] [4b – 2]}

The final answer is b{[4b + 2] [4b – 2]}

4. Given expression is 3a⁵ – 48a
Rewrite the given expression in the form of a² – b².
3a{(a²)² – (2²)²}
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a² and b = 2²
3a{[a² + 2²] [a² – 2²]}
From the above equation, [a² – 2²] is in the form of a² – b².
[(a)² – (2)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a and b = 2
[a + 2] [a – 2]
Now, 3a{[a² + 2²] [a² – 2²]}
3a{[a² + 4] [a + 2] [a – 2]}

The final answer is 3a{[a² + 4] [a + 2] [a – 2]}

5. Given expression is (3a – 4b)² – 25c²
Rewrite the given expression in the form of a² – b².
(3a – 4b)² – (5c)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 3a – 4b and b = 5c
{[3a – 4b + 5c] [3a – 4b – 5c]}

The final answer is {[3a – 4b + 5c] [3a – 4b – 5c]}

6. Given expression is (2m + 3n)² – 1
Rewrite the given expression in the form of a² – b².
(2m + 3n)² – (1)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 2m + 3n and b = 1
{[2m + 3n + 1] [2m + 3n – 1]}

The final answer is {[2m + 3n + 1] [2m + 3n – 1]}

7. Given expression is 16z² – (5x + y)²
Rewrite the given expression in the form of a² – b².
(4z)² – (5x + y)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 4z and b = 5x + y
{[4z + 5x + y] [4z – (5x + y)]}
{[4z + 5x + y] [4z – 5x – y]}

The final answer is {[4z + 5x + y] [4z – 5x – y]}

8. Given expression is 100 – (a – 5)²
Rewrite the given expression in the form of a² – b².
(10)² – (a – 5)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 10 and b = a – 5
{[10 + a – 5] [10 – (a – 5)]}
{[10 + a – 5] [10 – a + 5]}

The final answer is {[10 + a – 5] [10 – a + 5]}

9. (i) Given expression is (13)² – (12)²
Rewrite the given expression in the form of a² – b².
(13)² – (12)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 13 and b = 12
{[13 + 12] [13 – 12)]}
{[25] [1]} = 25

The final answer is 25

9. (ii) Given expression is (6.3)² – (4.2)²
Rewrite the given expression in the form of a² – b².
(6.3)² – (4.2)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 6.3 and b = 4.2
{[6.3 + 4.2] [6.3 – 4.2)]}
{[10.5] [2.1]} = 22.05

The final answer is 22.05

1. Given expression is a² + 8a + 16
The given expression a² + 8a + 16 is in the form a² + 2ab + b².
So find the factors of given expression using a² + 2ab + b² = (a + b)² = (a + b) (a + b) where a = a, b = 4
Apply the formula and substitute the a and b values.
a² + 8a + 16
(a)² + 2 (a) (4) + (4)²
(a + 4)²
(a + 4) (a + 4)

Factors of the a² + 8a + 16 are (a + 4) (a + 4)

2. Given expression is a² + 14a + 49
The given expression a² + 14a + 49 is in the form a² + 2ab + b².
So find the factors of given expression using a² + 2ab + b² = (a + b)² = (a + b) (a + b) where a = a, b = 7
Apply the formula and substitute the a and b values.
a² + 14a + 49
(a)² + 2 (a) (7) + (7)²
(a + 7)²
(a + 7) (a + 7)

Factors of the a² + 14a + 49 are (a + 7) (a + 7)

3. Given expression is 1 + 2a + a²
The given expression a² + 2a + 1 is in the form a² + 2ab + b².
So find the factors of given expression using a² + 2ab + b² = (a + b)² = (a + b) (a + b) where a = a, b = 1
Apply the formula and substitute the a and b values.
a² + 2a + 1
(a)² + 2 (a) (1) + (1)²
(a + 1)²
(a + 1) (a + 1)

Factors of the 1 + 2a + a² are (a + 1) (a + 1)

4. Given expression is 9 + 6c + c²
The given expression c² + 6c + 9 is in the form a² + 2ab + b².
So find the factors of given expression using a² + 2ab + b² = (a + b)² = (a + b) (a + b) where a = c, b = 3
Apply the formula and substitute the a and b values.
c² + 6c + 9
(c)² + 2 (a) (3) + (3)²
(c + 3)²
(c + 3) (c + 3)

Factors of the 9 + 6c + c² are (c + 3) (c + 3)

5. Given expression is m² + 6am + 9a²
The given expression m² + 6am + 9a² is in the form a² + 2ab + b².
So find the factors of given expression using a² + 2ab + b² = (a + b)² = (a + b) (a + b) where a = m, b = 3a
Apply the formula and substitute the a and b values.
m² + 6am + 9a²
(m)² + 2 (m) (3a) + (3a)²
(m + 3a)²
(m + 3a) (m + 3a)

Factors of the m² + 6am + 9a² are (m + 3a) (m + 3a)

6. Given expression is 4a² + 20a +25
The given expression 4a² + 20a +25 is in the form a² + 2ab + b².
So find the factors of given expression using a² + 2ab + b² = (a + b)² = (a + b) (a + b) where a = 2a, b = 5
Apply the formula and substitute the a and b values.
4a² + 20a +25
(2a)² + 2 (2a) (5) + (5)²
(2a + 5)²
(2a + 5) (2a + 5)

Factors of the 4a² + 20a +25 are (2a + 5) (2a + 5)

7. Given expression is 36a² + 36a + 9
The given expression 36a² + 36a + 9 is in the form a² + 2ab + b².
So find the factors of given expression using a² + 2ab + b² = (a + b)² = (a + b) (a + b) where a = 6a, b = 3
Apply the formula and substitute the a and b values.
36a² + 36a + 9
(6a)² + 2 (6a) (3) + (3)²
(6a + 3)²
(6a + 3) (6a + 3)

Factors of the 36a² + 36a + 9 are (6a + 3) (6a + 3)

8. Given expression is 9a² + 24a + 16
The given expression 9a² + 24a + 16 is in the form a² + 2ab + b².
So find the factors of given expression using a² + 2ab + b² = (a + b)² = (a + b) (a + b) where a = 3a, b = 4
Apply the formula and substitute the a and b values.
9a² + 24a + 16
(3a)² + 2 (3a) (4) + (4)²
(3a + 4)²
(3a + 4) (3a + 4)

Factors of the 9a² + 24a + 16 are (3a + 4) (3a + 4)

9. Given expression is a² + a + 1/4
The given expression a² + a + 1/4 is in the form a² + 2ab + b².
So find the factors of given expression using a² + 2ab + b² = (a + b)² = (a + b) (a + b) where a = a, b = 1/2
Apply the formula and substitute the a and b values.
a² + a + 1/4
(a)² + 2 (a) (1/2) + (1/2)²
(a + 1/2)²
(a + 1/2) (a + 1/2)

Factors of the a² + a + 1/4 are (a + 1/2) (a + 1/2)

10. The given expression a² – 6a + 9 is in the form a² – 2ab + b².
So find the factors of given expression using a² – 2ab + b² = (a – b)² = (a – b) (a – b) where a = a, b = 3
Apply the formula and substitute the a and b values.
a² – 6a + 9
(a)² – 2 (a) (3) + (3)²
(a – 3)²
(a – 3) (a – 3)

Factors of the a² – 6a + 9 are (a – 3) (a – 3)

1. The given expression a² – 10a + 25 is in the form a² – 2ab + b².
So find the factors of given expression using a² – 2ab + b² = (a – b)² = (a – b) (a – b) where a = a, b = 5
Apply the formula and substitute the a and b values.
a² – 10a + 25
(a)² – 2 (a) (5) + (5)²
(a – 5)²
(a – 5) (a – 5)

Factors of the a² – 10a + 25 are (a – 5) (a – 5)

2. The given expression 9a² – 12a + 4 is in the form a² – 2ab + b².
So find the factors of given expression using a² – 2ab + b² = (a – b)² = (a – b) (a – b) where a = 3a, b = 2
Apply the formula and substitute the a and b values.
9a² – 12a + 4
(3a)² – 2 (3a) (2) + (2)²
(3a – 2)²
(3a – 2) (3a – 2)

Factors of the 9a² – 12a + 4 are (3a – 2) (3a – 2)

3. The given expression 16a² – 24a + 9 is in the form a² – 2ab + b².
So find the factors of given expression using a² – 2ab + b² = (a – b)² = (a – b) (a – b) where a = 4a, b = 3
Apply the formula and substitute the a and b values.
16a² – 24a + 9
(4a)² – 2 (4a) (3) + (3)²
(4a – 3)²
(4a – 3) (4a – 3)

Factors of the 16a² – 24a + 9 are (4a – 3) (4a – 3)

4. The given expression 1 – 2a + a² is in the form a² – 2ab + b².
So find the factors of given expression using a² – 2ab + b² = (a – b)² = (a – b) (a – b) where a = a, b = 1
Apply the formula and substitute the a and b values.
1 – 2a + a²
(a)² – 2 (a) (1) + (1)²
(a – 1)²
(a – 1) (a – 1)

Factors of the 1 – 2a + a² are (a – 1) (a – 1)

5. The given expression 1 – 6a + 9a² is in the form a² – 2ab + b².
So find the factors of given expression using a² – 2ab + b² = (a – b)² = (a – b) (a – b) where a = 3a, b = 1
Apply the formula and substitute the a and b values.
1 – 6a + 9a²
(3a)² – 2 (3a) (1) + (1)²
(3a – 1)²
(3a – 1) (3a – 1)

Factors of the 1 – 6a + 9a² are (3a – 1) (3a – 1)

6. The given expression x²y² – 6xyz + 9z² is in the form a² – 2ab + b².
So find the factors of given expression using a² – 2ab + b² = (a – b)² = (a – b) (a – b) where a = xy, b = 3z
Apply the formula and substitute the a and b values.
x²y² – 6xyz + 9z²
(xy)² – 2 (xy) (3z) + (3z)²
(xy – 3z)²
(xy – 3z) (xy – 3z)

Factors of the x²y² – 6xyz + 9z² are (xy – 3z) (xy – 3z)

7. The given expression a² – 4ab + 4b² is in the form a² – 2ab + b².
So find the factors of given expression using a² – 2ab + b² = (a – b)² = (a – b) (a – b) where a = a, b = 2b
Apply the formula and substitute the a and b values.
a² – 4ab + 4b²
(a)² – 2 (a) (2b) + (2b)²
(a – 2b)²
(a – 2b) (a – 2b)

Factors of the a² – 4ab + 4b² are (a – 2b) (a – 2b)

(i) The given expression 2a² + 11a + 12
Rewrite the equation
2a² + 8a + 3a + 12
Group the first two terms and last two terms.
The first two terms are 2a² + 8a and the second two terms are 3a + 12.
Take 2a common from the first two terms.
2a (a + 4)
Take 3 common from the second two terms.
3 (a + 4)
2a (a + 4) + 3 (a + 4)
Then, take (a + 4) common from the above expression.
(a + 4) (2a + 3)

The final answer is (a + 4) (2a + 3).

(ii) The given expression 3a² + 8a + 4
Rewrite the equation
3a² + 6a + 2a + 4
Group the first two terms and last two terms.
The first two terms are 3a² + 6a and the second two terms are 2a + 4.
Take 3a common from the first two terms.
3a (a + 2)
Take 2 common from the second two terms.
2 (a + 2)
3a (a + 2) + 2 (a + 2)
Then, take (a + 2) common from the above expression.
(a + 2) (3a + 2)

The final answer is (a + 2) (3a + 2).

(iii) The given expression 3a² – 13a + 14
Rewrite the equation
3a² – 6a – 7a + 14
Group the first two terms and last two terms.
The first two terms are 3a² – 6a and the second two terms are – 7a + 14.
Take 3a common from the first two terms.
3a (a – 2)
Take -7 common from the second two terms.
-7 (a – 2)
3a (a – 2) – 7 (a – 2)
Then, take (a – 2) common from the above expression.
(a – 2) (3a – 7)

The final answer is (a – 2) (3a – 7).

(iv) The given expression 4a² – 7a + 3
Rewrite the equation
4a² – 4a – 3a + 3
Group the first two terms and last two terms.
The first two terms are 4a² – 4a and the second two terms are – 3a + 3.
Take 4a common from the first two terms.
4a (a – 1)
Take -3 common from the second two terms.
-3 (a – 1)
4a (a – 1) – 3 (a – 1)
Then, take (a – 1) common from the above expression.
(a – 1) (4a – 3)

The final answer is (a – 1) (4a – 3).

(v) The given expression 5a² – 11a – 12
Rewrite the equation
5a² – 15a + 4a – 12
Group the first two terms and last two terms.
The first two terms are 5a² – 15a and the second two terms are 4a – 12.
Take 4a common from the first two terms.
5a (a – 3)
Take 4 common from the second two terms.
4 (a – 3)
5a (a – 3) + 4 (a – 3)
Then, take (a – 3) common from the above expression.
(a – 3) (5a + 4)

The final answer is (a – 3) (5a + 4).

(vi) The given expression 7a² – 15a – 18
Rewrite the equation
7a² – 21a + 6a – 18
Group the first two terms and last two terms.
The first two terms are 7a² – 21a  and the second two terms are 6a – 18.
Take 7a common from the first two terms.
7a (a – 3)
Take 6 common from the second two terms.
6 (a – 3)
7a (a – 3) + 6 (a – 3)
Then, take (a – 3) common from the above expression.
(a – 3) (7a + 6)

The final answer is (a – 3) (7a + 6).

(i) The given expression 7a² + 6a – 1
Rewrite the equation
7a² + 7a – a – 1
Group the first two terms and last two terms.
The first two terms are 7a² + 7a and the second two terms are – a – 1.
Take 7a common from the first two terms.
7a (a + 1)
Take -1 common from the second two terms.
-1 (a + 1)
7a (a + 1) – 1 (a + 1)
Then, take (a + 1) common from the above expression.
(a + 1) (7a – 1)

The final answer is (a + 1) (7a – 1).

(ii) The given expression 9a² + 35a – 4
Rewrite the equation
9a² + 36a – a – 4
Group the first two terms and last two terms.
The first two terms are 9a² + 36a and the second two terms are – a – 4.
Take 9a common from the first two terms.
9a (a + 4)
Take -1 common from the second two terms.
-1 (a + 4)
9a (a + 4) – 1 (a + 4)
Then, take (a + 4) common from the above expression.
(a + 4) (9a – 1)

The final answer is (a + 4) (9a – 1).

(iii) The given expression 2a² – 5a + 3
Rewrite the equation
2a² – 2a – 3a + 3
Group the first two terms and last two terms.
The first two terms are 2a² – 2a and the second two terms are – 3a + 3.
Take 2a common from the first two terms.
2a (a – 1)
Take -3 common from the second two terms.
-3 (a – 1)
2a (a – 1) – 3 (a – 1)
Then, take (a – 1) common from the above expression.
(a – 1) (2a – 3)

The final answer is (a – 1) (2a – 3).

(iv) The given expression 7a – 6 – 2a²
Rewrite the equation
– 2a² + 4a + 3a – 6
Group the first two terms and last two terms.
The first two terms are – 2a² + 4a and the second two terms are 3a – 6.
Take -2a common from the first two terms.
-2a (a – 2)
Take 3 common from the second two terms.
3 (a – 2)
-2a (a – 2) + 3 (a – 2)
Then, take (a – 2) common from the above expression.
(a – 2) (-2a + 3)

The final answer is (a – 2) (-2a + 3).

(v) The given expression 11a² – 54a + 63
Rewrite the equation
11a² – 33a – 21a + 63
Group the first two terms and last two terms.
The first two terms are 11a² – 33a and the second two terms are – 21a + 63.
Take 11a common from the first two terms.
11a (a – 3)
Take -21 common from the second two terms.
-21 (a – 3)
11a (a – 3) – 21 (a – 3)
Then, take (a – 3) common from the above expression.
(a – 3) (11a – 21)

The final answer is (a – 3) (11a – 21).

(vi) The given expression a² + 2a – 3
Rewrite the equation
a² + 3a – a – 3
Group the first two terms and last two terms.
The first two terms are a² + 3a and the second two terms are – a – 3.
Take a common from the first two terms.
a (a + 3)
Take -1 common from the second two terms.
-1 (a + 3)
a (a + 3) – 1 (a + 3)
Then, take (a + 3) common from the above expression.
(a + 3) (a – 1)

The final answer is (a + 3) (a – 1).

(i) The given expression 2x² + 5x + 2
Rewrite the equation
2x² + 4x + x + 2
Group the first two terms and last two terms.
The first two terms are 2x² + 4x and the second two terms are x + 2.
Take 2x common from the first two terms.
2x (x + 2)
Take 1 common from the second two terms.
1 (x + 2)
2x (x + 2) + 1 (x + 2)
Then, take (x + 2) common from the above expression.
(x + 2) (2x + 1)

The final answer is (x + 2) (2x + 1).

(ii) The given expression 3a² + 14a + 8
Rewrite the equation
3a² + 12a + 2a + 8
Group the first two terms and last two terms.
The first two terms are 3a² + 12a and the second two terms are 2a + 8.
Take 3a common from the first two terms.
3a (a + 4)
Take 2 common from the second two terms.
2 (a + 4)
3a (a + 4) + 2 (a + 4)
Then, take (a + 4) common from the above expression.
(a + 4) (3a + 2)

The final answer is (a + 4) (3a + 2).

(iii) The given expression 2x² + 7x + 6
Rewrite the equation
2x² + 4x + 3x + 6
Group the first two terms and last two terms.
The first two terms are 2x² + 4x and the second two terms are 3x + 6.
Take 2x common from the first two terms.
2x (x + 2)
Take 3 common from the second two terms.
3 (x + 2)
2x (x + 2) + 3 (x + 2)
Then, take (x + 2) common from the above expression.
(x + 2) (2x + 3)

The final answer is (x + 2) (2x + 3).

(iv) The given expression 6a² – a – 15
Rewrite the equation
6a² + 9a -10a  – 15
Group the first two terms and last two terms.
The first two terms are 6a² + 9a and the second two terms are -10a  – 15.
Take 3a common from the first two terms.
3a (2a + 3)
Take -5 common from the second two terms.
-5 (2a + 3)
3a (2a + 3) -5 (2a + 3)
Then, take (2a + 3) common from the above expression.
(2a + 3) (3a – 5)

The final answer is (2a + 3) (3a – 5).

(v) The given expression 9s² – s – 8
Rewrite the equation
9s² – 9s + 8s – 8
Group the first two terms and last two terms.
The first two terms are 9s² – 9s and the second two terms are 8s – 8.
Take 9s common from the first two terms.
9s (s – 1)
Take 8 common from the second two terms.
8 (s – 1)
9s (s – 1) + 8 (s – 1)
Then, take (s – 1) common from the above expression.
(s – 1) (9s + 8)

The final answer is (s – 1) (9s + 8).

(vi) The given expression 12 + a – 6a²
Rewrite the equation
– 6a² + 9a -8a + 12
Group the first two terms and last two terms.
The first two terms are – 6a² + 9a and the second two terms are -8a + 12 .
Take -3a common from the first two terms.
-3a (2a – 3)
Take -4 common from the second two terms.
-4 (2a – 3)
-3a (2a – 3) – 4 (2a – 3)
Then, take (2a – 3) common from the above expression.
(2a – 3) (-3a – 4)

The final answer is (2a – 3) (-3a – 4).

(vii) The given expression 6 + 5x – 6x²
Rewrite the equation
– 6x² + 9x – 4x +6
Group the first two terms and last two terms.
The first two terms are – 6x² + 9x and the second two terms are – 4x +6.
Take -3x common from the first two terms.
-3x (2x – 3)
Take -2 common from the second two terms.
-2 (2x – 3)
-3x (2x – 3) -2 (2x – 3)
Then, take (2x – 3) common from the above expression.
(2x – 3) (-3x – 2)

The final answer is (2x – 3) (-3x – 2).

(viii) The given expression a² + 8a – 105
Rewrite the equation
a² + 15a – 7a – 105
Group the first two terms and last two terms.
The first two terms are a² + 15a and the second two terms are – 7a – 105.
Take a common from the first two terms.
a (a + 15)
Take -7 common from the second two terms.
-7 (a + 15)
a (a + 15) – 7 (a + 15)
Then, take (a + 15) common from the above expression.
(a + 15) (a – 7)

The final answer is (a + 15) (a – 7).

(i) The given expression is 3(a + 2) + 7(a + 2).
Factor out the common terms from the above expression. That is,
(a + 2) (3 + 7)
10 (a + 2).

The final answer is 10 (a + 2).

(ii) The given expression is (a + 3)a + (a + 3)4.
Factor out the common term from the above expression. That is,
(a + 3) (a + 4).

The final answer is (a + 3) (a + 4).

(iii) The given expression is 2(5a + 3b) + c(5a + 3b).
Factor out the common term from the above expression. That is,
(5a + 3b) (2 + c).

The final answer is (5a + 3b) (2 + c).

(iv) The given expression is 3a(b – 4c) – 5d(b – 4c).
Factor out the common term from the above expression. That is,
(b – 4c) (3a – 5d).

The final answer is (b – 4c) (3a – 5d).

(v) The given expression is x(p – q) + y(q – p).
x(p – q) – y(- q + p).
x(p – q) – y(p – q).
Factor out the common terms from the above expression. That is,
(p – q) (x – y).

The final answer is (p – q) (x – y).

(i) The given expression is p (q + r) – s (q + r).
Factor out the common terms from the above expression. That is,
(q + r) (p – s).

The final answer is (q + r) (p – s).

(ii) The given expression is 12(ab + 1) + 3x (ab + 1).
Factor out the common terms from the above expression. That is,
(ab + 1) (12 + 3x).

The final answer is (ab + 1) (12 + 3x).

(iii) The given expression is x^2 + y^2 + 9p(x^2 + y^2).
Factor out the common terms from the above expression. That is,
(x^2 + y^2) (1 + 9p).

The final answer is (x^2 + y^2) (1 + 9p).

(iv) The given expression is 3(x + y) – 5(x + y)^2.
Factor out the common terms from the above expression. That is,
(x + y) (3 – 5(x + y)).
(x + y) (3 – 5x – 5y).

The final answer is (x + y) (3 – 5x – 5y).

(i) The given expression is x (3y – 7z) – z (3y – 7z).
Factor out the common terms from the above expression. That is,
(3y – 7z) (x – z).

The final answer is (3y – 7z) (x – z).

(ii) The given expression is (2x – 6) (3p – 2q) – (2x – 6) (2q – 3p).
(2x – 6) (3p – 2q) – (2x – 6) ( – ) (3p – 2q).
(2x – 6) (3p – 2q) + (2x – 6) (3p – 2q).
Factor out the common terms from the above expression. That is,
2(2x – 6) (3p – 2q).

The final answer is 2(2x – 6) (3p – 2q).

(iii) The given expression is a(a + b) + (5a + 5b).
a(a + b) + 5(a + b).
Factor out the common terms from the above expression. That is,
(a + b) (a + 5).

The final answer is (a + b) (a + 5).

(iv) The given expression is (6ab + 3a) + (2b + 1).
3a(2b + 1) + (2b + 1).
Factor out the common terms from the above expression. That is,
(2b + 1) (3a + 1).

The final answer is (2b + 1) (3a + 1).

(v) The given expression is a(b – c)^2 – d(c – b)^3.
a(b – c)^2 + d(b – c)^3.
Factor out the common terms from the above expression. That is,
(b – c)^2 [a + d(b – c)].

The final answer is (b – c)^2 [a + d(b – c)].

(vi) The given expression is (a – 3) + (3xy – xya).
(a – 3) + xy( 3 – a).
(a – 3) – xy (a – 3).
Factor out the common terms from the above expression. That is,
(a – 3) (1 – xy).

The final answer is (a – 3) (1 – xy).

(i) The given expression is 12p + 15.
Expand the given expression. That is,
(3 * 4)p + (3 * 5).
Factor out the greatest common factor. That is,
3(4p + 5).

Then, 12p + 15 is equal to 3(4p + 5).

(ii) The given expression is 14p – 21.
Expand the given expression. That is,
(7 * 2)p – (7 * 3).
Factor out the greatest common factor. That is,
7(2p – 3).

Then, 14p – 21 is equal to 7(2p – 3).

(iii) The given expression is 9q – 12q^2.
Expand the given expression. That is,
(3 * 3)q – (3 * 4)q^2.
Factor out the greatest common factor. That is,
3q(3 – 4q).

Then, 9q – 12q^2 is equal to 3q(3 – 4q).

(i) The given expression is 16x^2 – 24xy.
Expand the given expression. That is,
(4 * 4)x^2 – (4 * 6)xy.
Factor out the greatest common factor. That is,
4x(4x – 6y).

Then, 16x^2 – 24xy is equal to 4x(4x – 6y).

(ii) The given expression is 15xy^2 – 20x^2y.
Expand the given expression. That is,
(5 * 3)xy^2 – (5 * 4)x^2y.
Factor out the greatest common factor. That is,
5xy(3y – 4x).

Then, 15xy^2 – 20x^2y is equal to 5xy(3y – 4x).

(iii) The given expression is 12a^2b^3 – 21a^3b^2.
Expand the given expression. That is,
(3 * 4)a^2b^3 – (3 * 7)a^3b^2.
Factor out the greatest common factor. That is,
3a^2b^2(4b – 7a).

Then, 12a^2b^3 – 21a^3b^2 is equal to 3a^2b^2(4b – 7a).

(i) The given expression is 24a^3 – 36a^2b.
Expand the given expression. That is,
(6 * 4)a^3 – (6 * 6)a^2b.
Factor out the greatest common factor. That is,
6a^2(4a – 6b).

Then, 24a^3 – 36a^2b is equal to 6a^2(4a – 6b).

(ii) The given expression is 10a^3 – 15a^2.
Expand the given expression. That is,
(5 * 2)a^3 – (5 * 3)a^2.
Factor out the greatest common factor. That is,
5a^2(2a – 3).

Then, 10a^3 – 15a^2 is equal to 5a^2(2a – 3).

(iii) The given expression is 36a^3b – 60a^2b^3c.
Expand the given expression. That is,
(6 * 6)a^3b – (6 * 10)a^2b^3c.
Factor out the greatest common factor. That is,
6a^2b(6a – 10b^2c).

Then, 36a^3b – 60a^2b^3c is equal to 6a^2b(6a – 10b^2c).

(i) The given expression is 9a³ – 6a² + 12a.
Expand the given expression. That is,
(3 * 3)a^3 – (3 * 2)a^2 + (3 * 4)a.
Factor out the greatest common factor. That is,
3a(3a^2 – 2a + 4).

Then, 9a^3 – 6a^2 + 12a is equal to 3a(3a^2 – 2a + 4).

(ii) The given expression is 8a^2 – 72ab + 12a.
Expand the given expression. That is,
(4 * 2)a^2 – (4 * 18)ab + (4 * 3)a.
Factor out the greatest common factor. That is,
4a(2a – 18b + 3a).

Then, 8a^2 – 72ab + 12a is equal to 4a(2a – 18b + 3a).

(iii) The given expression is 18x³y³ – 27x²y³ + 36x³y²
Expand the given expression. That is,
(9 * 2)x^3y^3 – (9 * 3)x^2y^3 + (9 * 4)x^3y^2.
Factor out the greatest common factor. That is,
9x^2y^2(2xy -3y + 4x).

Then, 18x³y³ – 27x²y³ + 36x³y² is equal to 9x^2y^2 (2xy -3y + 4x).

(i) The given expression is 14a^3 + 21a^4b – 28a^2b^2.
Expand the given expression. That is,
(7 * 2)a^3 + (7 * 3)a^4b – (7 * 4)a^2b^2.
Factor out the greatest common factor. That is,
7a^2(2a + 3a^2b – 4b^2).

Then, 14a^3 + 21a^4b – 28a^2b^2 is equal to 7a^2(2a + 3a^2b – 4b^2).

(ii) The given expression is -5 – 10x + 20x^2.
Expand the given expression. That is,
-5 – (5 * 2)x + (5 * 4)x^2.
Factor out the greatest common factor. That is,
5( -1 – 2x + 4x^2).

Then, -5 – 10x + 20x^2 is equal to 5( -1 – 2x + 4x^2) .

(i) The given expression is a(a + 3) + 5(a + 3).
Factor out the greatest common factor. That is,
(a + 3) (a + 5)

Then,a (a + 3) + 5(a + 3) is equal to (a + 3) (a + 5).

(ii) The given expression is 5a(a – 4) – 7(a – 4).
Factor out the greatest common factor. That is,
(a – 4) (5a – 7).

Then, 5a (a – 4) – 7 (a – 4) is equal to (a – 4) (5a – 7).

(iii) The given expression is 2x(1 – y) + 3(1 – y).
Factor out the greatest common factor. That is,
(1 – y) (2x + 3).

Then, 2x(1 – y) + 3(1- y) is equal to (1 – y) (2x + 3).

(i) The given expression is 6x (x – 2y) + 5y (x – 2y).
Factor out the greatest common factor. That is,
(x – 2y) (6x + 5y).

Then, 6x (x – 2y) + 5y (x – 2y) is equal to (x – 2y) (6x + 5y).

(ii) The given expression is p^3 (2x – y) + p^2 (2x – y).
Factor out the greatest common factor. That is,
(2x – y)p^2 (p + 1).

Then, p^3 (2x – y) + p^2 (2x – y) is equal to p^2(2x – y) (p + 1).

(i) The given expression is 9x (3x – 5y) – 12x²(3x – 5y).
Factor out the greatest common factor. That is,
3x(3x – 5y)(3 – 4x).

Then, 9x (3x – 5y) – 12x²(3x – 5y) is equal to 3x(3x – 5y)(3 – 4x).

(ii) The given expression is (a + 5)^2 – 4 (a + 5).
Factor out the greatest common factor. That is,
(a + 5) [(a + 5) – 4] = (a + 5) (a + 1).

Then, (a + 5)^2 – 4 (a + 5) is equal to (a + 5) (a + 1).

(iii) The given expression is 3(x – 2y)² – 5(x – 2y).
Factor out the greatest common factor. That is,
(x – 2y) [3(x – 2y) – 5].

Then, 3(x – 2y)² – 5(x – 2y) is equal to (x – 2y) [3(x – 2y) – 5].

(i) The given expression is 2x + 6y – 3(x + 3y)^2.
Factor out the greatest common factor. That is,
Factor out the greatest common factor. That is,
2(x + 3y) – 3(x + 3y)^2 = (x + 3y) (2 – 3(x + 3y)).

Then, 2x + 6y – 3(x + 3y)^2 is equal to (x + 3y) (2 – 3(x + 3y)).

(ii) The given expression is 16(2x – 3y)² – 4(2x – 3y).
Factor out the greatest common factor. That is,
4(2x – 3y) [4(2x – 3y) – 1].

Then, 16(2x – 3y)² – 4(2x – 3y) is equal to 4(2x – 3y) [4(2x – 3y) – 1].

(iii) The given expression is
p (x – 3) + q (3 – x).
we can write it as p (x – 3) – q(x – 3).
Factor out the greatest common factor. That is,
(x – 3) (p – q).

Then, p (x – 3) + q (3 – x) is equal to (x – 3) (p – q).

(iv) The given expression is 12 (2a – 3b)² – 16 (3b – 2a).
Expand the above expression. That is,
12 (2a – 3b)^2 + 32a – 48b.
We can write it as 12 (2a – 3b)^2 + 16 (2a – 3b).
Factor out the greatest common factor. That is,
4(2a – 3b) [3(2a – 3b) + 4].

Then, 12 (2a – 3b)² – 16 (3b – 2a) is equal to 4(2a – 3b) [3(2a – 3b) + 4].

(v) The given expression is (a + b)(2a + 5) – (a + b)(a + 3).
Factor out the greatest common factor. That is,
(a + b) [(2a + 5) – (a + 3)] = (a + b) (a + 2).

Then, (a + b)(2a + 5) – (a + b)(a + 3) is equal to (a + b) (a + 2).

(i) The given expression is xp + yp + xq + yq.
Factor out the greatest common factor. That is,
p(x + y) + q(x + y) = (x + y) (p + q).

Then, xp + yp + xq + yq is equal to (x + y) (p + q).

(ii) The given expression is p2 – sp – qp + sq.
Factor out the greatest common factor. That is,
p(p – s) – q(p – s) = (p – s) (p – q).

Then, p2 – sp – qp + sq is equal to (p – s) (p – q).

(iii) The given expression is xy² – yz² – xy + z²
Factor out the greatest common factor. That is,
xy^2 – xy – yz^2 + z^2 = xy(y – 1) – z(yz – z).

Then, xy² – yz² – xy + z² is equal to xy(y – 1) – z(yz – z).

(iv) : The given expression is a2 – ac + ab – bc.
Factor out the greatest common factor. That is,
a(a – c) + b(a – c) = (a – c) (a + b).

Then, a2 – ac + ab – bc is equal to (a – c) (a + b).

(v) The given expression is 6xy – y² + 12xz – 2yz.
Factor out the greatest common factor. That is,
y(6x – y) + 2z(6x – y) = (6x – y) (y + 2z).

Then, 6xy – y² + 12xz – 2yz is equal to (6x – y) (y + 2z).

(i) The given expression is (a – 2b)² + 4a–8b.
Factor out the greatest common factor. That is,
(a – 2b)² + 4(a – 2b) = (a – 2b) (a – 2b + 4).

Then, (a – 2b)² + 4a–8b is equal to (a – 2b) (a – 2b + 4).

(ii) The given expression is b² – ab (1 – a) – a³
we can write it as b^2 – ab + a^2b – a^3.
Factor out the greatest common factor. That is,
b(b – a) + a^2(b – a) = (b – a)(b + a^2).

Then, b² – ab (1 – a) – a³ is equal to (b – a)(b + a^2).

(iii) The given expression is (rp + sq)² + (sp – rq)²
We can write it as (rp)^2 + (sq)^2 + 2rpsq + (sp)^2 + (rq)^2 – 2rpsq
it is equal to (rp)^2 + (sq)^2 + (sp)^2 + (rq)^2
Factor out the greatest common factor. That is,
r^2(p^2 + q^2) + s^2(p^2 + q^2) = (p^2 + q^2) (r^2 + s^2)

Then, (rp + sq)² + (sp – rq)² is equal to (p^2 + q^2) (r^2 + s^2)

(iv) The given expression is pq² + (p – 1)q – 1.
we can write it as pq^2 + pq – q – 1= pq(q +1) – (q + 1)
Factor out the greatest common factor. That is,
(q + 1) (pq – 1).

Then, pq² + (p – 1)q – 1 is equal to (q + 1) (pq – 1).

(i) The given expression is p³ – 3p² + p – 3.
We can write it as p^2(p – 3) + (p – 3)
Factor out the greatest common factor. That is, (p – 3) (p^2 + 1).

Then, p³ – 3p² + p – 3is equal to (p – 3) (p^2 + 1).

(ii) The given expression is xy (p² + q²) – pq (x² + y²).
We can write it as xyp^2 + xyq^2 – pqx^2 – pqy^2.
Factor out the greatest common factor. That is, xp (yp –qx) – yq (yp–qx) = (yp – qx) (xp – yq).

Then, xy (p² + q²) – pq (x² + y²)is equal to (yp – qx) (xp – yq).

(iii) The given expression is p² – p(x + 2y) + 2xy.
We can write it as p^2 – px – 2py + 2xy.
Factor out the greatest common factor. That is, p(p -2y) – x(p – 2y) = (p – 2y) (p – x).

Then, p² – p(x + 2y) + 2xy is equal to (p – 2y) (p – x).

(i) Given expression is a² + ab + 9a + 9b.
Rearrange the terms
a² + ab + 9a + 9b.
Group the first two terms and last two terms.
The first two terms are a² + ab and the second two terms are 9a + 9b.
Take a common from the first two terms.
a (a + b)
Take 9 common from the second two terms.
9 (a + b)
a (a + b) + 9 (a + b)
Then, take (a + b) common from the above expression.
(a + b) (a + 9)

The final answer is (a + b) (a + 9).

(ii) Given expression is 6ab – 4b + 6 – 9a.
Rearrange the terms
6ab – 9a – 4b + 6
Group the first two terms and last two terms.
The first two terms are 6ab – 9a and the second two terms are – 4b + 6.
Take 3a common from the first two terms.
3a (2b – 3)
Take -2 common from the second two terms.
-2 (2b  – 3)
3a (2b – 3) – 2 (2b – 3)
Then, take (2b – 3) common from the above expression.
(2b – 3) (3a – 2)

The final answer is (2b – 3) (3a – 2).

(iii) Given expression is 10xy + 6x + 5y +3.
Rearrange the terms
10xy + 5y + 6x +3
Group the first two terms and last two terms.
The first two terms are 10xy + 5y and the second two terms are 6x +3.
Take 5y common from the first two terms.
5y (2x + 1)
Take 3 common from the second two terms.
3 (2x  + 1)
5y (2x + 1) + 3 (2x  + 1)
Then, take (2x  + 1) common from the above expression.
(2x  + 1) (5y + 3)

The final answer is (2x  + 1) (5y + 3).

(iv) Given expression is a³ + a² + a + 1.
Group the first two terms and last two terms.
The first two terms are a³ + a² and the second two terms are a + 1.
Take a² common from the first two terms.
a² (a + 1)
Take 1 common from the second two terms.
1 (a + 1)
a² (a + 1) + 1 (a + 1)
Then, take (a + 1) common from the above expression.
(a + 1) (a² + 1)

The final answer is (a + 1) (a² + 1).

(v) Given expression is m² – b + mb – m.
Rearrange the terms
m² + mb – b  – m.
Group the first two terms and last two terms.
The first two terms are m² + mb and the second two terms are – b – m.
Take m common from the first two terms.
m (m + b)
Take -1 common from the second two terms.
-1 (m + b)
m (m + b) -1 (m + b)
Then, take (m + b) common from the above expression.
(m + b) (m – 1)

The final answer is (m + b) (m – 1).

(vi) Given expression is a³ – a²b + 5a – 5b.
Rearrange the terms
a³ + 5a – a²b – 5b.
Group the first two terms and last two terms.
The first two terms are a³ + 5a and the second two terms are – a²b – 5b.
Take a common from the first two terms.
a (a² + 5)
Take -b common from the second two terms.
– b (a² + 5)
a (a² + 5) – b (a² + 5)
Then, take (m + b) common from the above expression.
(a² + 5) (a – b)

The final answer is (a² + 5) (a – b).

(vii) Given expression is x (x + 3) – x – 3.
Rearrange the terms
x²  – x + 3x – 3.
Group the first two terms and last two terms.
The first two terms are x²  – x  and the second two terms are 3x – 3.
Take x common from the first two terms.
x (x – 1)
Take 3 common from the second two terms.
3 (x – 1)
x (x – 1) + 3 (x – 1)
Then, take (x – 1) common from the above expression.
(x – 1) (x + 3)

The final answer is (x – 1) (x + 3).

(viii) Given expression is 3mx + 3my – 2nx – 2ny.
Rearrange the terms
3mx – 2nx + 3my – 2ny.
Group the first two terms and last two terms.
The first two terms are 3mx – 2nx  and the second two terms are 3my – 2ny.
Take x common from the first two terms.
x (3m – 2n)
Take y common from the second two terms.
y (3m – 2n)
x (3m – 2n) + y (3m – 2n)
Then, take (3m – 2n) common from the above expression.
(3m – 2n) (x + y)

The final answer is (3m – 2n) (x + y).

(ix) Given expression is a (a + b – c ) – bc.
Rearrange the terms
a² – ac + ab – bc.
Group the first two terms and last two terms.
The first two terms are a² – ac and the second two terms are ab – bc.
Take a common from the first two terms.
a (a – c)
Take b common from the second two terms.
b (a – c)
a (a – c) + b (a – c)
Then, take (a – c) common from the above expression.
(a – c) (a + b)

The final answer is (a – c) (a + b).

(x) Given expression is x³ – x² + xy² – x²y².
Rearrange the terms
x³ – x² – x²y² + xy²
Group the first two terms and last two terms.
The first two terms are x³ – x² and the second two terms are – x²y² + xy².
Take x² common from the first two terms.
x² (x – 1)
Take – xy² common from the second two terms.
– xy² (x – 1)
x² (x – 1) – xy² (x – 1)
Then, take (x – 1) common from the above expression.
(x – 1) (x² – xy²)

The final answer is (x – 1) (x² – xy²).

(i) Given expression is 4ab – 7b + 12a – 21.
Rearrange the terms
4ab + 12a – 7b – 21
Group the first two terms and last two terms.
The first two terms are 4ab + 12a and the second two terms are – 7b – 21.
Take 4a common from the first two terms.
4a (b + 3)
Take – 7 common from the second two terms.
– 7 (b + 3)
4a (b + 3) – 7 (b + 3)
Then, take (b + 3) common from the above expression.
(b + 3) (4a – 7)

The final answer is (b + 3) (4a – 7).

(ii) Given expression is 7xy – 5x – 28y + 20.
Rearrange the terms
7xy – 28y – 5x + 20
Group the first two terms and last two terms.
The first two terms are 7xy – 28y and the second two terms are – 5x + 20.
Take 7y common from the first two terms.
7y (x – 4)
Take – 5 common from the second two terms.
– 5 (x – 4)
7y (x – 4) – 5 (x – 4)
Then, take (x – 4) common from the above expression.
(x – 4) (7y – 5)

The final answer is (x – 4) (7y – 5).

(iii) Given expression is 5mn – 2m – 5n² + 2n.
Rearrange the terms
5mn – 5n² – 2m + 2n
Group the first two terms and last two terms.
The first two terms are 5mn – 5n² and the second two terms are – 2m + 2n.
Take n common from the first two terms.
5n (m – n)
Take – 2 common from the second two terms.
– 2 (m – n)
5n (m – n) – 2 (m – n)
Then, take (m – n) common from the above expression.
(m – n) (5n – 2)

The final answer is (m – n) (5n – 2).

(iv) Given expression is 6a² – 15ac – 8ba + 20bc.
Rearrange the terms
6a² – 8ba – 15ac + 20bc
Group the first two terms and last two terms.
The first two terms are 6a² – 8ba and the second two terms are – 15ac + 20bc.
Take 2a common from the first two terms.
2a (3a – 4b)
Take – 5c common from the second two terms.
– 5c (3a – 4b)
2a (3a – 4b) – 5c (3a – 4b)
Then, take (3a – 4b) common from the above expression.
(3a – 4b) (2a – 5c)

The final answer is (3a – 4b) (2a – 5c).

(v) Given expression is 4mx + 5nx – 12my – 15ny.
Rearrange the terms
4mx – 12my + 5nx – 15ny
Group the first two terms and last two terms.
The first two terms are 4mx – 12my and the second two terms are + 5nx – 15ny.
Take 4m common from the first two terms.
4m (x – 3y)
Take 5n common from the second two terms.
5n (x – 3y)
4m (x – 3y) + 5n (x – 3y)
Then, take (x – 3y) common from the above expression.
(x – 3y) (4m + 5n)

The final answer is (x – 3y) (4m + 5n).

(i) Given expression is 7mn – 21nr – 7mx + 21xr
Rearrange the terms
7mn – 7mx – 21nr + 21xr
Group the first two terms and last two terms.
The first two terms are 7mn – 7mx and the second two terms are – 21nr + 21xr.
Take 7m common from the first two terms.
7m (n – x)
Take -21r common from the second two terms.
-21r (n – x)
7m (n – x) – 21r (n – x)
Then, take (n – x) common from the above expression.
(n – x) (7m – 21r)

The final answer is (n – x) (7m – 21r).

(ii) Given expression is a² + ab(b + 1) + b³
Rearrange the terms
a² + ab + ab² + b³
Group the first two terms and last two terms.
The first two terms are a² + ab and the second two terms are ab² + b³.
Take a common from the first two terms.
a (a + b)
Take b² common from the second two terms.
b² (a + b)
a (a + b) + b² (a + b)
Then, take (a + b) common from the above expression.
(a + b) (a + b²)

The final answer is (a + b) (a + b²).

(iii) Given expression is yx² – 2x(1 – y) – 4
Rearrange the terms
yx² + 2xy – 2x – 4
Group the first two terms and last two terms.
The first two terms are yx² + 2xy and the second two terms are – 2x – 4.
Take xy common from the first two terms.
xy (x + 2)
Take -2 common from the second two terms.
-2 (x + 2)
xy (x + 2) – 2 (x + 2)
Then, take (x + 2) common from the above expression.
(x + 2) (xy – 2)

The final answer is (x + 2) (xy – 2).

(iv) Given expression is m² – m(a + 4b) + 4ab
Rearrange the terms
m² – 4mb – ma + 4ab
Group the first two terms and last two terms.
The first two terms are m² – 4mb and the second two terms are – ma + 4ab.
Take m common from the first two terms.
m (m – 4b)
Take -a common from the second two terms.
-a (m – 4b)
m (m – 4b) – a (m – 4b)
Then, take (m – 4b) common from the above expression.
(m – 4b) (m – a)

The final answer is (m – 4b) (m – a).

(v) Given expression is a – 9 – (a – 9)² + ab – 9b
Rearrange the terms
a – 9 – (a – 9)² + ab – 9b
Group the first two terms and last two terms.
The first two terms are a – 9 – (a – 9)²  and the second two terms are ab – 9b.
Take m common from the first two terms.
(a – 9) (1 – a + 9) = (a – 9) (10 – a)
Take b common from the second two terms.
b (a – 9)
(a – 9) (10 – a) + b (a – 9)
Then, take (a – 9) common from the above expression.
(a – 9) (10 – a + b)

The final answer is (a – 9) (10 – a + b).

(i) Given expression is (a – 4) – (a – 4)² + 12 – 3a
Rearrange the terms
(a – 4) – (a – 4)² – 3a + 12
Group the first two terms and last two terms.
The first two terms are (a – 4) – (a – 4)²  and the second two terms are – 3a + 12.
Take (a – 4) common from the first two terms.
(a – 4) (1 – a + 4) = (a – 4) (5 – a)
Take -3 common from the second two terms.
-3 (a – 4)
(a – 4) (5 – a) – 3 (a – 4)
Then, take (a – 4) common from the above expression.
(a – 4) (5 – a – 3) = (a – 4) (2 – a)

The final answer is (a – 4) (2 – a).

(ii) Given expression is b (c – d )² – a (d – c) + 3c – 3d
Rearrange the terms
b (c – d )² + a (c – d) + 3c – 3d
Group the first two terms and last two terms.
The first two terms are b (c – d )² + a (c – d) and the second two terms are 3c – 3d.
Take (c – d) common from the first two terms.
(c – d) (b (c – d) + a) = (c – d) (bc – bd + a)
Take 3 common from the second two terms.
3 (c – d)
(c – d) (bc – bd + a) + 3 (c – d)
Then, take (a – 4) common from the above expression.
(c – d) (bc – bd + a + 3)

The final answer is (c – d) (bc – bd + a + 3).

(iii) Given expression is (a² + 2a)² – 7 (a² + 2a) – y (a² + 2a) +7y
Rearrange the terms
(a² + 2a)² – y (a² + 2a) – 7 (a² + 2a) + 7y
Group the first two terms and last two terms.
The first two terms are (a² + 2a)² – y (a² + 2a) and the second two terms are – 7 (a² + 2a) + 7b.
Take (a² + 2a) common from the first two terms.
(a² + 2a) (a² + 2a – y)
Take – 7 common from the second two terms.
– 7 (a² + 2a – y)
(a² + 2a) (a² + 2a – y) – 7 (a² + 2a – y)
Then, take (a² + 2a – y) common from the above expression.
(a² + 2a – y) (a² + 2a – 7)

The final answer is (a² + 2a – y) (a² + 2a – 7).

(iv) Given expression is m⁴x + m³ (2x – y ) – m(2my + z) – 2z
Rearrange the terms
m⁴x + 2xm³ – ym³ –  2m²y – zm – 2z
Group the first two terms, middle terms, and last two terms.
The first two terms are m⁴x + 2xm³, the middle terms are – ym³ –  2m²y, and the second two terms are – zm – 2z.
Take xm³ common from the first two terms.
xm³ (m + 2)
Take – m²y common from the middle two terms.
– m²y (m + 2)
Take – z common from the middle two terms.
– z (m + 2)
xm³ (m + 2) – m²y (m + 2) – z (m + 2)
Then, take (m + 2) common from the above expression.
(m + 2) (xm³ – m²y – z)

The final answer is (m + 2) (xm³ – m²y – z).

(v) Given expression is x³ – 2x²y + 3xy² – 6y³
Rearrange the terms
x³ – 2x²y + 3xy² – 6y³
Group the first two terms, and last two terms.
The first two terms are x³ – 2x²y, and the second two terms are 3xy² – 6y³.
Take x² common from the first two terms.
x² (x – 2y)
Take 3y² common from the middle two terms.
3y² (x – 2y)
x² (x – 2y) + 3y² (x – 2y)
Then, take (x – 2y) common from the above expression.
(x – 2y) (x² + 3y²)

The final answer is (x – 2y) (x² + 3y²).

(vi) Given expression is m² + n – mn – m
Rearrange the terms
m² – m + n – mn
Group the first two terms, and last two terms.
The first two terms are m² – m, and the second two terms are n – mn.
Take m common from the first two terms.
m (m – 1)
Take -n common from the middle two terms.
-n (m – 1)
m (m – 1) – n (m – 1)
Then, take (m – 1) common from the above expression.
(m – 1) (m – n)

The final answer is (m – 1) (m – n).

(vii) Given expression is 5ab – b² + 15ca – 3bc
Rearrange the terms
5ab + 15ca – b² – 3bc
Group the first two terms, and last two terms.
The first two terms are 5ab + 15ca, and the second two terms are – b² – 3bc.
Take 5a common from the first two terms.
5a (b + 3c)
Take -b common from the middle two terms.
-b (b + 3c)
5a (b + 3c) – b (b + 3c)
Then, take (b + 3c) common from the above expression.
(b + 3c) (5a – b)

The final answer is (b + 3c) (5a – b).

(viii) Given expression is xy² – yz² – xy + z²
Rearrange the terms
xy² – xy – yz² + z²
Group the first two terms, and last two terms.
The first two terms are xy² – xy, and the second two terms are – yz² + z².
Take xy common from the first two terms.
xy (y – 1)
Take -z² common from the middle two terms.
-z² (y – 1)
xy (y – 1) – z² (y – 1)
Then, take (y – 1) common from the above expression.
(y – 1) (xy – z²)

The final answer is (y – 1) (xy – z²).

(i) Given expression is a² – 9
Rewrite the given expression in the form of a² – b².
(a)² – (3)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a and b = 3
[a + 3] [a – 3]

The final answer is [a + 3] [a – 3]

(ii) Given expression is x² – 1
Rewrite the given expression in the form of a² – b².
(x)² – (1)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = x and b = 1
[x + 1] [x – 1]

The final answer is [x + 1] [x – 1]

(iii) Given expression is 49 – a²
Rewrite the given expression in the form of a² – b².
(7)² – (a)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 7 and b = a
[7 + a] [7 – a]

The final answer is [7 + a] [7 – a]

(iv) Given expression is 4a² – 25
Rewrite the given expression in the form of a² – b².
(2a)² – (5)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 2a and b = 5
[2a + 5] [2a – 5]

The final answer is [2a + 5] [2a – 5]

(v) Given expression is x²y² – 16
Rewrite the given expression in the form of a² – b².
(xy)² – (4)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = xy and b = 4
[xy + 4] [xy – 4]

The final answer is [xy + 4] [xy – 4]

(vi) Given expression is m⁴ – n⁴
Rewrite the given expression in the form of a² – b².
(m²)² – (n²)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = m² and b = n²
[m² + n²] [m² – n²]
From the above equation, [m² – n²] is in the form of a² – b².
[(m)² – (n)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = m and b = n
[m + n] [m – n]
Now, [m² + n²] [m² – n²]
[m² + n²] [m + n] [m – n]

The final answer is [m² + n²] [m + n] [m – n]

(i) Given expression is 144x² – 169y²
Rewrite the given expression in the form of a² – b².
(12x)² – (13y)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 12x and b = 13y
[12x + 13y] [12x – 13y]

The final answer is [12x + 13y] [12x – 13y]

(ii) Given expression is 1 – 0.09x²
Rewrite the given expression in the form of a² – b².
(1)² – (0.3x)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 1 and b = 0.3x
[1 + 0.3x] [1 – 0.3x]

The final answer is [1 + 0.3x] [1 – 0.3x]

(iii) Given expression is 16m² – 121
Rewrite the given expression in the form of a² – b².
(4m)² – (11)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 4m and b = 11
[4m + 11] [4m – 11]

The final answer is [4m + 11] [4m – 11]

(iv) Given expression is – 64x² + (9/25) y²
Rewrite the given expression in the form of a² – b².
(9/25) y² – 64x² = (3/5y)² – (8x)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 3/5y and b = 8x
[3/5y + 8x] [3/5y – 8x]

The final answer is [3/5y + 8x] [3/5y – 8x]

(v) Given expression is m⁴ – 256
Rewrite the given expression in the form of a² – b².
(m²)² – ( (4)²)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = m² and b = (4)²
[m² + (4)²] [m² – (4)²]
[m² + 16] [(m)² – (4)²]
From the above equation, [(m)² – (4)²] is in the form of a² – b².
[(m)² – (4)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = m and b = (4)
[m + (4)] [m – (4)]
Now, [m² + 16] [(m)² – (4)²]
[m² + 16] [m + (4)] [m – (4)]
[m² + 16] [m + 4] [m – 4]

The final answer is [m² + 16] [m + 4] [m – 4]

(vi) Given expression is (a + b)⁴ – c⁴
Rewrite the given expression in the form of a² – b².
((a + b)²)² – ( (c)²)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = (a + b)² and b = ((c)²)²
[(a + b)² + (c)²] [(a + b)² – (c)²]
[(a + b)² + (c)²] [(a + b)² – (c)²]
From the above equation, [(a + b)² – (c)²] is in the form of a² – b².
[(a + b)² – (c)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a + b and b = (c)
[a+ b + c] [a + b – c]
Now, [(a + b)² + (c)²] [(a + b)² – (c)²]
[(a + b)² + (c)²] [a+ b + c] [a + b – c]

The final answer is [(a + b)² + (c)²] [a+ b + c] [a + b – c]

(i) Given expression is 36x² – y²
Rewrite the given expression in the form of a² – b².
(6x)² – (y)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 6x and b = (y)
[6x + y] [6x – y]

The final answer is [6x + y] [6x – y]

(ii) Given expression is a²b² – 16
Rewrite the given expression in the form of a² – b².
(ab)² – (4)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = ab and b = 4
[ab + 4] [ab – 4]

The final answer is [ab + 4] [ab – 4]

(iii) Given expression is 9x⁴y⁴ – 25m⁴n⁴
Rewrite the given expression in the form of a² – b².
(3x²y²)² – (5m²n²)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 3x²y² and b = 5m²n²
[3x²y² + 5m²n²] [3x²y² – 5m²n²]

The final answer is [3x²y² + 5m²n²] [3x²y² – 5m²n²]

(iv) Given expression is a⁴ – 256
Rewrite the given expression in the form of a² – b².
(a²)² – ((4)²)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a² and b = (4)²
[a² + (4)²] [a² – (4)²]
[a² + 16] [a² – (4)²]
From the above equation, [a² – (4)²] is in the form of a² – b².
[(a)² – (4)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a and b = 4
[a + 4] [a – 4]
Now, [a² + 16] [a² – (4)²]
[a² + 16] [a + 4] [a – 4]

The final answer is [a² + 16] [a + 4] [a – 4]

(v) Given expression is 81m² – 49n²
Rewrite the given expression in the form of a² – b².
(9m)² – (7n)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 9m and b = 7n
[9m + 7n] [9m – 7n]

The final answer is [9m + 7n] [9m – 7n]

(vi) Given expression is a² – (b – c)²
Rewrite the given expression in the form of a² – b².
(a)² – (b – c)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a and b = b – c
[a + b – c] [a – (b – c)]
[a + b – c] [a – b + c]

The final answer is [a + b – c] [a – b + c]

(i) Given expression is 16a² – (3b + 2y)²
Rewrite the given expression in the form of a² – b².
(4a)² – (3b + 2y)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 4a and b = 3b + 2y
[4a + 3b + 2y] [4a – (3b + 2y)]
[4a + 3b + 2y] [4a – 3b – 2y]

The final answer is [4a + 3b + 2y] [4a – 3b – 2y]

(ii) Given expression is (3m + 4n)² – (4n + 5n)²
Rewrite the given expression in the form of a² – b².
(3m + 4n)² – (9n)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 3m + 4n and b = 9n
[3m + 4n + 9n] [3m + 4n – 9n]
[3m + 13n] [3m – 5n]

The final answer is [3m + 13n] [3m – 5n]

(iii) Given expression is (a + b)² – (a – b)²
Rewrite the given expression in the form of a² – b².
(a + b)² – (a – b)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a + b and b = a – b
[a + b + a – b] [a + b – (a – b)]
[2a] [a + b – a + b] = 2a (2b) = 4ab

The final answer is 4ab

(iv) Given expression is 50x² – 72y²
Rewrite the given expression in the form of a² – b².
2[(5x)² – (6y)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 5x and b = 6y
2{[5x + 6y] [5x – 6y]}

The final answer is 2{[5x + 6y] [5x – 6y]}

(v) Given expression is x⁴ – (y + z)⁴
Rewrite the given expression in the form of a² – b².
[(x²)² – ((y + z)²)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = x² and b = (y + z)²
{[x² + (y + z)²] [x² – (y + z)²]}
{[x² + y² + z²+ 2yz] [x² – (y + z)²]}
From the above equation, [x² – (y + z)²] is in the form of a² – b².
[x² – (y + z)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = x and b = y + z
[x + y + z] [x – (y + z)] = [x + y + z] [x – y – z]
Now, {[x² + y² + z²+ 2yz] [x² – (y + z)²]}
{[x² + y² + z²+ 2yz] [x + y + z] [x – y – z]}

The final answer is {[x² + y² + z²+ 2yz] [x + y + z] [x – y – z]}

(vi) Given expression is x² – 1/169
Rewrite the given expression in the form of a² – b².
(x)² – (1/13)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = x and b = 1/13
[x + 1/13] [x – 1/13]

The final answer is [x + 1/13] [x – 1/13]

(i) Given expression is9 (a + b)² – 4 (a – b)²
Rewrite the given expression in the form of a² – b².
(a + b)² – (2 (a – b))²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a + b and b = 2 (a – b)
[a + b + 2 (a – b)] [a + b – 2 (a – b)]
[a + b + 2a – 2b] [a + b – 2a + 2b]
[3a – b] [3b – a]

The final answer is [3a – b] [3b – a]

(ii) Given expression is 16/49 – 25m²
Rewrite the given expression in the form of a² – b².
(4/7)² – (5m)²
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 4/7 and b = 5m
[4/7 + 5m] [4/7 – 5m]

The final answer is [4/7 + 5m] [4/7 – 5m]

(iii) Given expression is 9ab² – a³
Rewrite the given expression in the form of a² – b².
a[(3b)² – (a)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 3b and b = a
a{[3b + a] [3b – a]}

The final answer is a{[3b + a] [3b – a]}

(iv) Given expression is 4 (3a + 1)² – 9 (a – 2)²
Rewrite the given expression in the form of a² – b².
4 (9a² + 1 + 6a) – 9 (a² + 4 – 4a) = 36a² + 4 + 24a – 9a² – 36 + 36a = 27a² + 60a – 32 = (9a – 4) (3a + 8)
Now, apply the formula of a² – b² = (a + b) (a – b),
(9a – 4) (3a + 8)

The final answer is (9a – 4) (3a + 8)

(v) Given expression is 1 – 121m²
Rewrite the given expression in the form of a² – b².
[(1)² – (11m)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 1 and b = 11m
{[1 + 11m] [1 – 11m]}

The final answer is {[1 + 11m] [1 – 11m]}

(vi) Given expression is 169x² – 1
Rewrite the given expression in the form of a² – b².
[(13x)² – (1)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 13x and b = 1
{[13x + 1] [13x – 1]}

The final answer is {[13x + 1] [13x – 1]}

(i) Given expression is 1 – (m + n)²
Rewrite the given expression in the form of a² – b².
[(1)² – (m + n)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 1 and b = m + n
{[1 + m + n] [1 – (m + n)]}
{[1 + m + n] [1 – m – n]}

The final answer is {[1 + m + n] [1 – m – n]}

(ii) Given expression is a²b² – 25/c²
Rewrite the given expression in the form of a² – b².
[(ab)² – (5/c)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = ab and b = 5/c
{[ab + 5/c] [ab – (5/c)]}
{[ab + 5/c] [ab – 5/c]}

The final answer is {[ab + 5/c] [ab – 5/c]}

(iii) Given expression is a¹²b⁴ – a⁴b¹²
Rewrite the given expression in the form of a² – b².
[(a⁶b²)² – (a²b⁶)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a⁶b² and b = a²b⁶
{[a⁶b² + a²b⁶] [a⁶b² – a²b⁶]} = a²b²{[a⁴ + b⁴] a²b²[a⁴ – b⁴]} = a⁴b⁴ [a⁴ + b⁴] [a⁴ – b⁴]
From the above equation, [(a²)² – (b²)²] is in the form of a² – b².
[(a²)² – (b²)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a² and b = b²
[a² + b²] [a² – b²] 
Now, a⁴b⁴ [a⁴ + b⁴] [a⁴ – b⁴]
a⁴b⁴ [a⁴ + b⁴] [a² + b²] [a² – b²]
From the above equation, [a² – b²] is in the form of a² – b².
[a² – b²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a and b = b
[a + b] [a – b]
Now, a⁴b⁴ [a⁴ + b⁴] [a² + b²] [a² – b²]
a⁴b⁴ [a⁴ + b⁴] [a² + b²] [a + b] [a – b]

The final answer is a⁴b⁴ [a⁴ + b⁴] [a² + b²] [a + b] [a – b]

(iv) Given expression is 100 (m – n)² – 121 (x + y)²
Rewrite the given expression in the form of a² – b².
[(10(m – n))² – (11 (x + y))²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 10(m – n) and b = 11 (x + y)
{[10(m – n) + 11 (x + y)] [10(m – n) – 11 (x + y)]}
{[10m – 10n + 11x + 11y] [10m – 10n – 11x – 11y]}

The final answer is {[10m – 10n + 11x + 11y] [10m – 10n – 11x – 11y]}

(v) Given expression is 2a – 50a³
Rewrite the given expression in the form of a² – b².
2a[(1)² – (5a)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 1 and b = 5a
2a {[1 + 5a] [1 – 5a]}

The final answer is 2a {[1 + 5a] [1 – 5a]}

(vi) Given expression is 25/a² – (4a²)/9
Rewrite the given expression in the form of a² – b².
[(5/a)² – (2a/3)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 5/a and b = 2a/3
{[5/a + 2a/3] [5/a – 2a/3]}

The final answer is {[5/a + 2a/3] [5/a – 2a/3]}

(vii) Given expression is a⁴ – 1/(b⁴)
Rewrite the given expression in the form of a² – b².
[(a²)² – (1/b²)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a² and b = 1/b²
{[a² + 1/b²] [a² – 1/b²]}
From the above equation, [a² – 1/b²] is in the form of a² – b².
[a² – (1/b)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = a and b = 1/b
[a + 1/b] [a – 1/b]
Now, {[a² + 1/b²] [a² – 1/b²]}
{[a² + 1/b²] [a + 1/b] [a – 1/b] }

The final answer is {[a² + 1/b²] [a + 1/b] [a – 1/b]}

(vi) Given expression is 75a³b² – 108ab⁴
Rewrite the given expression in the form of a² – b².
3ab²[(5a)² – (6b)²]
Now, apply the formula of a² – b² = (a + b) (a – b), where a = 5a and b = 6b
3ab²{[5a + 6b] [5a – 6b]}

The final answer is 3ab²{[5a + 6b] [5a – 6b]}

(i) The given expression is 3x + 21
Here, the first term is 3x and the second term is 21
By comparing the above two terms, we can observe the greatest common factor and that is 3
Now, factor out the greatest common factor from the expression
That is, 3 [x + 7]
3 [x + 7]

Therefore, the resultant value for the expression 3x + 21 is 3 [x + 7]

(ii) The given expression is 7a – 14
Here, the first term is 7a and the second term is 14
By comparing the above two terms, we can observe the greatest common factor and that is 7
Now, factor out the greatest common factor from the expression
That is, 7 [a – 2]
7 [a – 2]

Therefore, the resultant value for the expression 7a – 14 is 7 [a – 2]

(iii) The given expression is b³ + 3b
Here, the first term is b³ and the second term is 3b
By comparing the above two terms, we can observe the greatest common factor and that is b
Now, factor out the greatest common factor from the expression
That is, b [b² + 3]
b [b² + 3]

Therefore, the resultant value for the expression b³ + 3b is b [b² + 3]

(iv) The given expression is 20a + 5a²
Here, the first term is 20a and the second term is 5a²
By comparing the above two terms, we can observe the greatest common factor and that is 5a
Now, factor out the greatest common factor from the expression
That is, 5a [4 + a]
5a [4 + a]

Therefore, the resultant value for the expression 20a + 5a² is 5a [4 + a]

(v) The given expression is – 16m + 20m³
Here, the first term is – 16m, and the second term is 20m³
By comparing the above two terms, we can observe the greatest common factor and that is 4m
Now, factor out the greatest common factor from the expression
That is, 4m [-4 + 5m²]
4m [-4 + 5m²]

Therefore, the resultant value for the expression – 16m + 20m³ is 4m [-4 + 5m²]

(vi) The given expression is 5a²b + 15ab²
Here, the first term is 5a²b and the second term is 15ab²
By comparing the above two terms, we can observe the greatest common factor and that is 5ab
Now, factor out the greatest common factor from the expression
That is, 5ab [a + 3b]
5ab [a + 3b]

Therefore, the resultant value for the expression 5a²b + 15ab² is 5ab [a + 3b]

(vii) The given expression is 9m² + 5m
Here, the first term is 9m² and the second term is 5m
By comparing the above two terms, we can observe the greatest common factor and that is m
Now, factor out the greatest common factor from the expression
That is, m [9m + 5]
m [9m + 5]

Therefore, the resultant value for the expression 9m² + 5m is m [9m + 5]

(viii) The given expression is 19x – 57y
Here, the first term is 19x and the second term is – 57y
By comparing the above two terms, we can observe the greatest common factor and that is 19
Now, factor out the greatest common factor from the expression
That is, 19 [x – 3y]
19 [x – 3y]

Therefore, the resultant value for the expression 19x – 57y is 19 [x – 3y]

(ix) The given expression is 25x²y²z³ – 15xy³z
Here, the first term is 25x²y²z³ and the second term is – 15xy³z
By comparing the above two terms, we can observe the greatest common factor and that is 5xy²z
Now, factor out the greatest common factor from the expression
That is, 5xy²z [5xz² – 3y]
5xy²z [5xz² – 3y]

Therefore, the resultant value for the expression 25x²y²z³ – 15xy³z is 5xy²z [5xz² – 3y]

(i) The given expression is 13x + 39
Here, the first term is 13x and the second term is 39
By comparing the above two terms, we can observe the greatest common factor and that is 13
Now, factor out the greatest common factor from the expression
That is, 13 [x + 3]
13 [x + 3]

Therefore, the resultant value for the expression 13x + 39 is 13 [x + 3]

(ii) The given expression is 19a – 57b
Here, the first term is 19a and the second term is – 57b
By comparing the above two terms, we can observe the greatest common factor and that is 19
Now, factor out the greatest common factor from the expression
That is, 19 [a – 3b]
19 [a – 3b]

Therefore, the resultant value for the expression 19a – 57b is 19 [a – 3b]

(iii) The given expression is 21ab + 49abc
Here, the first term is 21ab and the second term is 49abc
By comparing the above two terms, we can observe the greatest common factor and that is 7ab
Now, factor out the greatest common factor from the expression
That is, 7ab [3 + 7c]
7ab [3 + 7c]

Therefore, the resultant value for the expression 21ab + 49abc is 7ab [3 + 7c]

(iv) The given expression is – 16x + 20x³
Here, the first term is – 16x and the second term is 20x³
By comparing the above two terms, we can observe the greatest common factor and that is 4x
Now, factor out the greatest common factor from the expression
That is, 4x [-4 + 5x²]
4x [-4 + 5x²]

Therefore, the resultant value for the expression – 16x + 20x³ is 4x [-4 + 5x²]

(v) The given expression is 12a²b – 42abc
Here, the first term is 12a²b and the second term is – 42abc
By comparing the above two terms, we can observe the greatest common factor and that is 6ab
Now, factor out the greatest common factor from the expression
That is, 6ab [2a – 7bc]
6ab [2a – 7bc]

Therefore, the resultant value for the expression 12a²b – 42abc is 6ab [2a – 7bc]

(vi) The given expression is 27m³n³ + 36m⁴n²
Here, the first term is 27m³n³ and the second term is 36m⁴n²
By comparing the above two terms, we can observe the greatest common factor and that is 9m³n²
Now, factor out the greatest common factor from the expression
That is, 9m³n² [3n + 4m]
9m³n² [3n + 4m]

Therefore, the resultant value for the expression 27m³n³ + 36m⁴n² is 9m³n² [3n + 4m]