How to write this as a multiplication: $4 x^2 y^2 - (x^2 + y^2 - z^2)^2$?

Question

I have the following polynomial:

$$ 4 x^2 y^2 - (x^2 + y^2 - z^2)^2$$

(which comes up, for example, in computing the area of a triangle using the cosine law).

I would like to convert this to a product. Wolfram tells me it's

$$ -(x - y - z) (x + y - z) (x - y + z) (x + y + z)$$

How can I find this form if I don't already know it? What operations should I perform?

Answer 1

$$ \begin{align} \color{magenta}{4 x^2 y^2 - (x^2 + y^2 - z^2)^2} & = \color{brown}{(2xy +x^2+y^2-z^2)(2xy -x^2-y^2+z^2)} \\ &= \color{blue}{((x+y)^2-z^2)(-(x-y)^2 -z^2)}\\ & = \boxed{\color{red}{-(x+y+z)(x+y-z)(x-y-z)(x-y+z)}} \end {align}$$

Answer 2

Note that you have a binomial formula:

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

Where $a=2xy$ since $a^2=4x^2y^2$ and $b=x^2+y^2-z^2$ obviously.

This leads to $4x^2y^2-(x^2+y^2-z^2)^2=(2x^2y^2-x^2-y^2+z^2)(2x^2y^2+x^2+y^2-z^2)$

With additional use of binomial formulas you are able to factor this term.