Let $f \in \mathbb{Z}[x]$ be a polynomial of degree at least $1$. Prove that there is $n \in \mathbb{Z} $ such that the corresponding polynomial function $f(n)$ is not a prime.
I think that the task is to show that there is always an $n$ for any degree polynomial, that would give me a non-prime output. I was thinking I could make a case distinction for even and odd $n$, but this argument does not get me anywhere. Does anybody have small hint what would be a good direction to explore? the book gave the hint "Many ways to attack this. Think of something." - which was really helpful, as you might imagine. $f(n)$ would have the form:
$$f(n)= a_0 + a_1 n + a_2 n^2 \dots $$