HINT: Use the fact that there exists a homomorphism of rings \(\displaystyle \phi: \mathbb{Z}[x] \rightarrow \mathbb{Z}_p[x]\) (you are just quotienting out the ideal generated by \(\displaystyle p\), \(\displaystyle p\mathbb{Z}\)).

Personally, I would do question (b), then extend it for all primes p. This is the contrapositive of (a)*, so you are done.

That is a bit roundabout though, so there may be an easier way...

*The contrapositive: Instead of proving \(\displaystyle A \Rightarrow B\) you prove \(\displaystyle not B \Rightarrow not A\). These two things are equivalent.