Given a quadratic polynomial $ax^2 + bx + c$, with $a$, $b$ and $c$ being integers, is there a characterization of all primes $p$ for which the equation $ax^2 + bx + c \equiv 0 \pmod p$ has solutions?
I have seen it mentioned that it follows from quadratic reciprocity that the set is precisely the primes in some arithmetic progression, but the statement may require some tweaking. The set of primes modulo which $1 + \lambda = \lambda^2$ has solutions seems to be $5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, \dots$ which are ($5$ and) the primes that are $1$ or $9$ modulo $10$.
(Can the question also be answered for equations of higher degree?)