Question

( x + 2 )^2 - ( y + 2 )^2

Use the difference of two squares identity to help you rewrite the expression in factored form. Be sure to simplify each factor.

Answer 1

The difference of two squares factoring pattern states that a difference of two squares can be factored as follows:

$a^2-b^2 = (a+b)(a-b)$

So, whenever you recognize the two terms of a subtraction to be two squares, you can factor it as the sum of the roots multiplied by the difference of the roots.

In this case, the squares are obvious: $(x+2)^2$ is the square of $x+2$, and $(y+2)^2$ is the square of $y+2$

So, we can factor the expression as

$(x+2)^2 - (y+2)^2 = [(x+2)+(y+2)] - [(x+2)+(y+2)]$

(the round parenthesis aren't necessary, I used them only to make clear the two terms)

We can simplify the expression summing like terms:

$(x+2)^2 - (y+2)^2 = [(x+2)+(y+2)][(x+2)-(y+2)] = (x+y+4)(x-y)$