Finding solutions of diophantine equation

Question

I'm trying to find the solutions of the general diophantine equation $x^3y^2 - 2x^4y + 2x^3 +y^2 - 2xy=0$. By trial, $x=0, y=0$ is a trivial solution but I'd like to see if there are other solutions, any idea on how to proceed is helpful. I tried grouping the variables and seeking a solution criterion based on divisability or bounds but with no success. Any guideline is useful.

Answer 1

$x^3y^2 - 2x^4y + 2x^3 +y^2 - 2xy=0$

You can write as -

$y^2(x^3 + 1) - 2xy (x^3+1) + 2x^3 + 2 = 2$

$(x^3 + 1) (y^2 - 2xy + 2) = 2$

There are only two factors $1$ and $2$. That gives you $x = y = 0$ and $x = y = 1$