Only finitely many primes dividing the values of an integer polynomial

Question

For a given $P\in \mathbb{Z}[x]$ call a positive prime $p$ good if there exists $n\in \mathbb{Z}$ such that $p$ divides $P(n)$. Does there exist a non-constant $P$ such that the set of good primes is finite?

Answer 1

The answer is no. Given any finite set of primes $S$, the number of positive integers up to $x$ divisible only by primes in $S$ grows like a constant times $(\log x)^{\#S}$. Since the number of values, up to $x$, of a degree-$d$ polynomial grows like a constant times $x^{1/d}$, those values must inevitably include integers with prime factors outside of $S$.