Rewrite $x^{4}-256$ as $\left(x^{2}\right)^{2}-16^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(x^{2}-16\right)\left(x^{2}+16\right)$$
Consider $x^{2}-16$. Rewrite $x^{2}-16$ as $x^{2}-4^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.
$$\left(x-4\right)\left(x+4\right)$$
Rewrite the complete factored expression. Polynomial $x^{2}+16$ is not factored since it does not have any rational roots.
