Extension of claim regarding polynomials yielding composite (as opposed to prime) values

Question

It is easy to prove that, if $p$ is any nonconstant polynomial with integer coefficients, then there is some positive integer $n$ such that $p(n)$ is composite. Is it also known to be true that there are infinitely many such $n$ for a given $p$?

Answer 1

Hint: If $d = p(n)$, then $d \mid p(n + kd) \; \forall \; k \in \mathbb{Z}$. To show this, use the binomial theorem to expand $p(n + kd)$, and then collect the terms to form $p(n)$. Note each of the remaining terms has at least one factor of $kd$.