My answer sheet says =(x^{2})^{2}-(2^{2})^{2}
= (x^{2}-2^{2})(x^{2}+2^{2})
=(x-2)(x+2)(x^{2}+2)
My question is, how did it break that down like that?
x^4 - 16
This is the difference of two squares, which factors to
(x^2 + 4)(x^2 - 4)
You can't factor x^2 + 4 anymore, but x^2 - 4 is another difference of squares, which can be factored to (x + 2)(x - 2), the final answer is
(x^2 + 4)(x + 2)(x - 2)