The factors of $x^4+64$ are: 
$\textbf{(A)}\ (x^2+8)^2\qquad\textbf{(B)}\ (x^2+8)(x^2-8)\qquad\textbf{(C)}\ (x^2+2x+4)(x^2-8x+16)\\ \textbf{(D)}\ (x^2-4x+8)(x^2-4x-8)\qquad\textbf{(E)}\ (x^2-4x+8)(x^2+4x+8)$
Notice that: 
\begin{align*} x^4+64  &= x^4+2^6 \\ 	&= x^4+4\cdot 2^4 \end{align*}
So, using the [Sophie Germain Identity](https://artofproblemsolving.com/wiki/index.php/Sophie_Germain_Identity), we can factor the expression as:
\begin{align*} x^4+64  & = x^4+4\cdot 2^4\\ 	& = (x^2+2\cdot2^2-4x)(x^2+2\cdot2^2+4x)\\         & = (x^2-4x+8)(x^2+4x+8) \end{align*}
So it is clear that our answer is $\boxed{\textbf{(E)}}$.