<s>[INST] $y^2+5xy+6x^2-9x-4y=0$ where $x$ and $y$ are integers Need some help solving this for integers $x$ and $y$:
$$
y^2+5xy+6x^2-9x-4y=0
$$
I managed to make something like this:
$$
(y+3x-4)(y+2x)=x\\
(y+3x)(y+2x-3)=y
$$
Find integers for $x$ and $y$ that satisfy the equations above.
But, what do I do next, or is this a bad approach?
 [/INST] Hint:  written as a quadratic in $\,y\,$, the discriminant of $\;y^2+(5x-4)y+3x(2x-3)\,$ must be a perfect square: $\;\Delta_y=(5x-4)^2-12x(2x-3)=x^2 - 4 x + 16=(x-2)^2+12\,$.
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