Solve $z^4+2z^3+3z^2+2z+1 =0$ Solve $z^4+2z^3+3z^2+2z+1 =0$ with $z$: a complex variable.
Attempt at solving the problem:
We divide the polynom by $z^2$ and we get:
$z^2+2z+3+\dfrac{2}{z}+  \dfrac{1}{z^2}=0  $  $         $ We set $w=z+  \dfrac{1}{z}$
We now have $w^2+2w+5=0$
$\bigtriangleup = -16$
Let's find $\omega$ such that $\omega^2=-16$
We have $\omega=4i$
Therefore we have the 2 roots:
$w_    {1}=-1-2i$ and $ w_    {2}=-1+2i  $
The issue is: I don't know how to find z


"I don't know how to find z"

Sure you do!
....
If you are correct in what you have done so far and you have
$z + \frac 1z = w$
And $w_1 = -1-2i$ and $w_2 = -1 + 2i$ then
you need to solve $z +\frac 1z = (-1-2i)$ or $z^2 +(1+2i)z + 1=0$
ANd $z + \frac 1z = (-1+2i)$ or $z^2 + (1-2i)z + 1 = 0$.
Both of which can be solved by quadratic equation to get the four solutions
$z = \frac {(-1\pm 2i) \pm\sqrt{(1\mp 2i)^2 -4}}{2}$
(And if you are worried about expressing $\sqrt{M}$ just convert to polar coordinates.)
That is.... if everything you've done is correct.
