Let $x, y$ be integers. If we factor $x+y$ can we avoid factoring $x-y$?

Question

Let $x + y = a b$ be a factorization of $x + y$. Since $x - y = x + y - 2y$, we have $x - y = a b - 2y$. If $a | 2y$ then $x - y = a(b - z)$ for some integer $z$. Hence, in this case only $x+y$ needed to be factored to obtain the factorization of $x - y$. In what other cases can a factorization of $x+y$ be used to quickly obtain the factorization of $x-y$?