<s>[INST] Find all positive integer solutions for the following equation: Find all positive integer solutions for the following equation:
$(x^2+1)(y^2+1)+2(x-y)(1-xy)=4(1+xy)$
I've tried simplifying the equation and then refactoring but I can't find any solutions.
 [/INST] Using wolfram alpha, all the solutions are given by the expression : $$\begin{cases}y=\frac{x-1}{x+1}, x\neq -1 \\y = \frac {x+3}{x+1}, x\neq -1 \end{cases}$$
For example, the solutions for $x=1$ are $y=0$ and $y=2$.
You could try to find these solutions by hand solving the quadratic equation in respect to $y$. If you expand the original equation you get:
$$x^2 y^2 - 2 x^2 y + x^2 + 2 x y^2 + 2 x + y^2 - 2 y + 1 = 4 x y + 4$$ $$\Leftrightarrow (x^2 +2x+1) y^2 + (-2x^2-4x-2)y + (x^2+2x+1-4)=0 $$
Which you can solve using the quadratic formula.
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