How many integers $n$ are there such that the quantity $\lvert 2n^2 + 23n + 11 \rvert$ is prime?

Question

How many integers $n$ are there such that the quantity $\lvert 2n^2 + 23n + 11 \rvert$ is prime?

I know this equation can be factored as $(2n+1)(n+11)$ and positive positive integer is prime when it's only factors are 1 and itself. Any hints are greatly appreciated.

Answer 1

Hint: for this factorization not to show that your number is not prime, at least one of $|2n+1|$ and $|n+11|$ must be $1$.