Question

How do you factor completely `a^2 - 4b^2 - 4a + 4`?

Answer 1

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

Explanation:

The difference of squares identity can be written:

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

We use this with `A=(a-2)` and `B=2b` as follows:

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

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

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

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