# Polynomial (quartic) with complex coefficient: root localization

For all $$u\in \mathbb C$$, let $$P_u(X)= X^4+4X+u$$. I know that the roots of $$P_u$$ could be found explicitly, but it seems to lead to inextricably complex calculations.

Let $$H= \{z \in \mathbb C ~|~ \mathrm{Re}(z)<1/2\}$$,

$$Q_a^+ = \{z \in \mathbb C ~|~ \mathrm{Re}(z)>a, \mathrm{Im}(z)>a\}$$,

$$Q_a^- = \{z \in \mathbb C ~|~ \mathrm{Re}(z)>a, \mathrm{Im}(z)<-a\}$$.

I would like to show that there exists $$a>1/2$$ such that for all $$u$$, $$P_u$$ always has at least one root in $$Q_a^+$$, one in $$Q_a^-$$ and one in $$H$$.

I found some interesting results about polynomial roots localization, but often they show some bound which don't seem to help here (even if I take the reciprocal polynomial).

Any help would be useful.

Here is an example of roots with $$u=1+I$$, and $$a=0.7$$.

This is false. For instance, if $$u=-1000$$, then according to WolframAlpha the roots are approximately $$-5.65$$, $$5.59$$, and $$0.03\pm 5.62i$$ and so none of them are in $$Q_a^+$$ or $$Q_a^-$$ for any $$a>1/2$$. As motivation for trying this example, note that when $$u$$ is large, you can expect the roots of $$P_u$$ to be close to the fourth roots of $$-u$$, so when $$-u$$ is large and positive the roots will be close to the axes and thus avoid $$Q_a^+$$ and $$Q_a^-$$. (In fact, $$-u=4$$ is already large enough.)
It is true that there is always a root in $$H$$, since the roots must add up to $$0$$ (since $$P_u$$ has no $$x^3$$ term) and in particular at least one must have nonpositive real part.