<s>[INST] Solve the following system: $(x^2+1)(y^2+1) = 12xy$ and $(x+1)^2(y+1)^2=30xy$ I have been given the following system to solve:
$$(x^2+1)(y^2+1) = 12xy$$
$$(x+1)^2(y+1)^2=30xy$$
I noticed that the system is symmetric, but any of the methods I know to solve symmetric systems doesn't seem right. I have also tried writing the second equation as $$((x^2+1)+2x)((y^2+1)+2y) = (x^2+1)(y^2+1)+2y(x^2+1)+2x(y^2+1)+4xy = 30xy$$,
but this didn't get me anywhere. I would appreciate any hints towards the right solution.
 [/INST] Hint
From a formal point of view, you face an octic polynomial.
Use the first equation and solve for $y$
$$y_\pm=\frac{6x\pm\sqrt{-x^4+34 x^2-1}}{x^2+1}$$ Using $y_+$, plug it in the second, group terms and factor; you should have
$$12 x \left(x^2-4 x+1\right) \left(x^2-3 x+1\right)+$$ $$2 \left(x^2-4 x+1\right) \left(x^2-3 x+1\right)\sqrt{-x^4+34 x^2-1}=0$$ that is to say
$$\left(x^2-4 x+1\right) \left(x^2-3 x+1\right)(6x+\sqrt{-x^4+34 x^2-1})=0$$
Now, it is simple.
Do the same using $y_-$.
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