I was told to assume $f(x)$ is a polynomial with degree $d\geq 1$ with integer coefficients and positive leading coefficient.
(i) I need to show that there are infinitely many $x$ such that $f(x)$ isn't prime.
(ii) I also need to show that if $f(x_0) = m$, where $m>0$, then $f(x) \equiv 0 \pmod m$ whenever $x\equiv x_0 \pmod m$.
I tried (ii) and so far I have
If $f(x_0) = m \neq 0 \pmod m$, we have $x \equiv x_0 \pmod m$ so then $f(x) \equiv 0 \pmod m$.
For (i) I am not sure where to start can someone give me an idea?