Consider $a^{2}b^{4}-64a^{2}$. Factor out $a^{2}$.
$$a^{2}\left(b^{4}-64\right)$$
Consider $b^{4}-64$. Rewrite $b^{4}-64$ as $\left(b^{2}\right)^{2}-8^{2}$. The difference of squares can be factored using the rule: $p^{2}-q^{2}=\left(p-q\right)\left(p+q\right)$.
$$\left(b^{2}-8\right)\left(b^{2}+8\right)$$
Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: $b^{2}-8,b^{2}+8$.
