I have to find the domain of
So, there can't be any $z$ that makes null the denominator,
$z^2+z+1$ = 0
and after decomposing $z$,
How to solve that quadratic complex equation?
The maximal domain of definition is,as you imply, the complement in $\mathbb C$ of the set of roots of the polynomial $z^2+z+1$.
Therefore to find the bad points, we need to solve the equation $$z^2+z+1=0.$$ I am pretty sure, now, that you know how to solve quadratic equations, no?