Q:

Solve for all $z$: $z^3+4|z|=0$ in a better way

I need to solve for all $z$
$$z^3+4|z|=0$$
It's quite straight forward, once you insert $a+bi$ instead of $z$.
the answers are:
$a=0$ and $b=0$
or $a=1$  and  $b=\sqrt{3}$ 
or $a=1$ and $b=-\sqrt{3}$ 
From experience, these exercises usually have some better, shorter solutions than this, does somebody have an idea?
It's important to note that these are my answers and not a textbook's so I might have a mistake, please correct if I'm wrong

A:

Since $$z^3=-4|z|\implies |z|^3 =4|z| \implies |z|\in\{0,2\}$$
we have two possibilities:
a) $|z|=0$ so $z=0$.
b) $|z|=2$ so $z^3+8=0$ so $$(z+2)(z^2-2z+4)=0$$
so $z=-2$ or $z=1\pm i\sqrt{3}$

