In section 13.6, there is this proof:
My question is about the highlighted portions. It is to my understanding that $\mathbb{Z}[x]$ is not an ED, and we cannot do division here. But how did the authors just assume that we have a factorization here? Is it because of Gauss' Lemma? (If so, how exactly was the logic applied?)
I am assuming that the two highlighted portions have the same reasoning, but correct me if I'm wrong.