Question

factorize : 6(a+b)^2+5(a^2-b^2)-6(a-b)^2

Answer 1

First factor $a^{2} - b^{2} to (a - b) * (a + b)$
Let (a + b) = x and (a - b) = y

Then we have $6 x^{2} + 5xy - 6 y^{2} \\ 6 x^{2} + 9xy - 4xy - 6 y^{2} \\ 3x(2x + 3y) - 2y(2x + 3y) \\ (3x - 2y) (2x + 3y)$

Now 3x - 2y -> 3(a + b) - 2 (a - b) -> a + 5b
and 2x + 3y -> 2(a + b) + 3(a - b) -> 5a - b

Answer
(a + 5b) * (5a - b)

Answer 2

$6(a+b)^2+5(a^2-b^2)-6(a-b)^2\\\\=6(a+b)^2-6(a-b)^2+5a^2-5b^2\\\\=6\underbrace{[(a+b)^2-(a-b)^2]}_{use\ x^2-y^2=(x-y)(x+y)}+5a^2-5b^2\\\\=6[a+b-(a-b)](a+b+a-b)+5a^2-5b^2\\\\=6(a+b-a+b)(2a)+5a^2-5b^2\\\\=6(2b)(2a)+5a^2-5b^2$

$=24ab+5a^2-5b^2\\\\=5a^2-ab+25ab-5b^2\\\\=a(5a-b)+5b(5a-b)\\\\=(5a-b)(a+5b)$