jmc

Begin by noticing that $256=16^2$. We can then repeatedly apply a difference of squares factorization: $$\begin{array}{rl}{x}^8-256& =(x^4+16)(x^4-16)\\ & =(x^4+16)(x^2+4)(x^2-4)\\ & =\boxed{(x^4+16)(x^2+4)(x+2)(x-2)}\end{array}$$