The question is, Given MCQ, Which of the following is true?

(a) $Z[x]$ is principal ideal domain.

(b) $Z[x,y]/\langle y+1\rangle$ is a unique factorization domain.

(c) If $R$ is a principal ideal domain and $p$ is a non-zero prime ideal, then $R/P$ has finitely many prime ideal.

(d) If $R$ is principal ideal domain, then any subring of $R$ containing $I$ is again a principle ideal domain.

The correct option are (b) and (c). I got the option (c) is correct. For option (b), it was written in the explanation, that $\frac{\mathbb{Z[x,y]}}{\langle y+1\rangle}\cong \mathbb{Z[x]}$ and since $\mathbb{Z[x]}$ is Unique Factorization Domain, $\frac{\mathbb{Z[x,y]}}{\langle y+1\rangle}$ is also unique factorization domain.

My question, how we got that $\frac{\mathbb{Z[x,y]}}{\langle y+1\rangle}\cong \mathbb{Z[x]}$? How just seeing this quotient ring, we get the idea that this quotient ring can be isomorphic to $\mathbb{Z[x]}$?