Solve $z^4 + 4 = 0$ I'm trying to solve it by using its polar form, but then I get
$$
\begin{align*}
z^4 &= (\rho e^{i\phi})^4 = \rho^4 e^{4i\phi}\\
    &= -4 = -4 e^{0i}\\
\end{align*}
$$
From the definition of equality of complex numbers, $\rho^4 = -4$ and $4\phi = 0 + 2\pi k$ for some $k \in \mathbb{Z}$. 
This would mean $\rho = \sqrt{-2}$ and $\phi = \pi k / 2$. I have no idea how to interpret that imaginary radius, and Wolfram Alpha says the angle is $\pi k / 4$. Should this equation be solved using this method? Am I missing some step, or are my calculations incorrect?
I've already read Solve $z^4+1=0$ algebraically, but I want to know whether this equation is solvable using this method or another method should be used.

Something that we all never learned in high school, but should have, is the amazing factorization $X^4+4=(X^2+2X+2)(X^2-2X+2)$. With this, the total factorization is easy.
