CMOQR 2022 Problem 4

For a non-negative integer $n$ , call a one-variable polynomial $F$ with integer coefficients $n$ -good if: (a) $F(0) = 1$ (b) For every positive integer $c$ , $F(c) > 0$ , and (c) There exist exactly $n$ values of $c$ such that $F(c)$ is prime. Show that there exist infinitely many non-constant polynomials that are not $n$ -good for any $n$ .