$P$ non-constant polynomial, then there is for each $a>0$ there is $b>0$ such that $$(z:\left | z\right |>b)\subseteq (P(z):\left | z \right |>a)$$
I'm not sure where to start or how to prove this statement, any help would be great.
Edit: I first wrote $b<0$, which is clearly false. And as corrected, $b>0$ is right one.
The $z$'s in your two sets play different roles, so it might be better to write this as
$$ \{w: |w| > b\} \subseteq \{P(z): |z| > a \} $$
$\{P(z): |z| \le a\}$ is a compact set, so it is bounded. Since the Fundamental Theorem of Calculus says $P$ is surjective, all you need to do is take $b$ so $\{P(z): |z| \le a\} \subseteq \{w: |w| \le b\}$, and then $ \{w: |w| > b\} \subseteq \{P(z): |z| > a \}$.