Does have at most 2 solutions for primes $p\geq 5$?
Mathematica confirms this for the first 10K primes:
Max[ Table[ Length[ Reduce[{ Mod[2*x^2 + 2*x + 3, Prime[i]] == 0, 0 <= x <= Prime[i]}, x,Integers]], {i,3,10000}] ] Output: 2
but I wasn't sure if it was true in general, and if it could be proved.
Inspired by: Elementary proof that $2x^2+xy+3y^2$ represents infinitely many primes?