Solve $e^{4z} +e^{3z} + e^{2z} + e^z + 1 = 0$. Solve $$e^{4z} +e^{3z} + e^{2z} + e^z  + 1 = 0.$$
I have attempted this problem with the usual definition by writing $z=x+iy$ and using $e^z = e^x(\cos y + i \sin y)$ but got a large unsolvable mess. Is there something I'm missing?

$e^{4z}+e^{3z}+e^{2z}+e^z+1=0$ is too hard to solve directly. Instead, solve
\begin{equation}
e^{5z}-1=(e^z-1)(e^{4z}+e^{3z}+e^{2z}+e^z+1)=0
\end{equation}
and remove roots that are not roots of original equation.
