smc

Notice that: $$\begin{array}{rl}{x}^4+64& =x^4+2^6\\ & =x^4+4\cdot 2^4\end{array}$$ So, using the Sophie Germain Identity, we can factor the expression as: $$\begin{array}{rl}{x}^4+64& =x^4+4\cdot 2^4\\ & =(x^2+2\cdot 2^2-4x)(x^2+2\cdot 2^2+4x)\\ & =(x^2-4x+8)(x^2+4x+8)\end{array}$$ So it is clear that our answer is $\boxed{(x^2-4x+8)(x^2+4x+8)}$.