Find the number of integers $n$ such that the equation $xy^2 + y^2 - x - y= n$ has an infinite number of integer solutions $(x,y)$.
Firstly the equation can be rearranged into $(y-1)(y+x(y+1)) = n$. If $n$ is $0$, there's an infinite number of $(x,y)$ possible. However, I am struggling with the case when $n$ is a nonzero integer. I had a peek at the solution, and it says
There exists a divisor $k$ of $n$ such that $y-1 = k$ and $x(y+1) + y = n/k$ for infinitely many $x$. It forces $y+1 = 0$.
Yet I don't quite see how $y+1=0$... If anyone can help me understand that'd be great!