Irreducible polynomials over $\mathbb{Z}_p$

Question

Let odd prime $p$ be given. Is there a way to determine the:

(1) number of irreducible polynomials $x^2+bx+1$ over $\mathbb{Z}_p$?

(2) possible values of $b$ for which $x^2+bx+1$ over $\mathbb{Z}_p$ is irreducible?

Thank you so much!