1. Factoring Completely

16 - 4x^4 - 4y^2 - 8x^2y

= (-4x^4 - 4y^2 - 8x^2y) + 16
= -4(x^4 + y^2 + 2x^2y) + 16
= -4(x^4+y^2 + 2x^2y) + 4^2
= 4^2 - [4(x^4 + y^2 + 2x^2y)]
= 4^2 - [2(x+y)]^2
= 4(2 + x^2 + y)(2 - x^2 - y)

What I don't understand is the transition from 4^2 - [2(x+y)]^2 to 4(2 + x^2 + y)(2 - x^2 - y)

If someone could explain it, that would be great thanks.

2. Originally Posted by lax600
16 - 4x^4 - 4y^2 - 8x^2y

= (-4x^4 - 4y^2 - 8x^2y) + 16
= -4(x^4 + y^2 + 2x^2y) + 16
= -4(x^4+y^2 + 2x^2y) + 4^2
= 4^2 - [4(x^4 + y^2 + 2x^2y)]

Mr F chips in:

$\displaystyle {\color{red}= 4[4 - (x^4 + y^2 + 2x^2 y)]}$

$\displaystyle {\color{red} = 4[2^2 - (x^2 + y)^2]}$

$\displaystyle {\color{red} = 4[A^2 - B^2]}$ where $\displaystyle {\color{red}A = 2}$ and $\displaystyle {\color{red}B = (x^2 + y)}$

= 4(2 + x^2 + y)(2 - x^2 - y)

using the difference of two squares formula.

What I don't understand is the transition from 4^2 - [2(x+y)]^2 to 4(2 + x^2 + y)(2 - x^2 - y)

If someone could explain it, that would be great thanks.
..

3. oh hehheh...