<s>[INST] Find all ordered pairs of real numbers $(x, y)$ such that $x^2y^2 + 2xy^2 + 5x^2 + 3y^2 + 10x + 5 = 0$. How can I find all ordered pairs of real numbers $(x, y)$ such that $x^2y^2 + 2xy^2 + 5x^2 + 3y^2 + 10x + 5 = 0$?
I don't see a pattern, I don't know how to proceed.  Any answer is greatly appreciated.
 [/INST] $$  y^2 \; ( x^2 + 2x + 3) = -5(x^2 + 2 x + 1) $$
$$  y^2 = -5 \; \; \frac{(x+1)^2 }{2 + (x+1)^2} $$
The right hand side is never positive. It is only zero when $x=-1$ 
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