Answer
$(x^2+2x+2)(x^2-2x+2)$
Work Step by Step
We factor $x^4+4$ by adding and subtracting $4x^2$, then using the difference of squares:
$$\begin{align*}
x^4+4&=x^4+4+4x^2-4x^2\\
&=(x^4+4x^2+4)-4x^2\\
&=(x^2+2)^2-(2x)^2\\
&=(x^2+2+2x)(x^2+2-2-2x)\\
&=(x^2+2x+2)(x^2-2x+2).
\end{align*}$$
