Answer
$z=-2$ or $z= \pm \sqrt{3}$
Work Step by Step
Rewrite the equation to equal zero, and factor by grouping.
$$
\begin{aligned}
z^3+2 z^2-3 z-6 & =0 \\
z^2(z+2)-3(z+2) & =0 \\
(z+2)\left(z^2-3\right) & =0
\end{aligned}
$$
So, $z+2=0$ or $z^2-3=0$, thus $z=-2$ or $z= \pm \sqrt{3}$.
