Question

How do you factor `(a-b)^2 - 16(a+2b)^2`?

Answer 1

`(a-b)^2-16(a+2b)^2 = -3(a+3b)(5a+7b)`

Explanation:

The difference of squares identity can be written:

`A^2-B^2=(A-B)(A+B)`

Let `A=(a-b)` and `B=4(a+2b)`

Then we find:

`(a-b)^2-16(a+2b)^2 = (a-b)^2-(4(a+2b))^2`

`color(white)((a-b)^2-16(a+2b)^2) = ((a-b)-4(a+2b))((a-b)+4(a+2b))`

`color(white)((a-b)^2-16(a+2b)^2) = (a-b-4a-8b)(a-b+4a+8b)`

`color(white)((a-b)^2-16(a+2b)^2) = (-3a-9b)(5a+7b)`

`color(white)((a-b)^2-16(a+2b)^2) = -3(a+3b)(5a+7b)`