Question

How do you factor completely `a^2 - 1 - 4b^2 - 4b`?

Answer 1

`a^2-1-4b^2-4b = (a-2b-1)(a+2b+1)`

Explanation:

The difference of squares identity can be written:

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

Use this with `A=a` and `B=2b+1` as follows:

`a^2-1-4b^2-4b = a^2-(4b^2+4b+1)`

`color(white)(a^2-1-4b^2-4b) = a^2-(2b+1)^2`

`color(white)(a^2-1-4b^2-4b) = (a-(2b+1))(a+(2b+1))`

`color(white)(a^2-1-4b^2-4b) = (a-2b-1)(a+2b+1)`