Putnam 1979 B3

Problem B-3 Let $F$ be a finite field having an odd number $m$ of elements. Let $p(x)$ be an irreducible (i.e., nonfactorable) polynomial over $F$ of the form $$x^{2}+b x+c, b, c \in F$$

For how many elements $k$ in $F$ is $p(x)+k$ irreducible over $F$ ?