Answer
$(x^4+16)(x^2+4)(x+2)(x-2)$
Work Step by Step
Using $x^2-y^2=(x+y)(x-y)$ or the factoring of the difference of 2 squares, then the factored form of the given expression, $
x^8-2^8
,$ is
\begin{array}{l}
(x^4+2^4)(x^4-2^4)
\\\\=
(x^4+2^4)(x^2+2^2)(x^2-2^2)
\\\\=
(x^4+2^4)(x^2+2^2)(x+2)(x-2)
\\\\=
(x^4+16)(x^2+4)(x+2)(x-2)
\end{array}