Elaboration on Correspondence Theorem Result

Can someone explain why $\mathbb{Z}[x]/(x^2+3,5) \cong \mathbb{F}_{25}$
$Z[x] \to Z_{5}[x]$ by f(x) $\in Z[x] = a_nx^n + ... + a_0 \to (a_n) mod 5 x^n +... + (a_0) mod 5$. is a homomorphism.
Then $Z[x]/ \to Z_{5}[x]/$ since x^2 + 3 is irreducible in $Z_{5}[x]$ (no zeros are there). $Z_{5}[x]/$ is a field containing $5^2$ elements.