<s>[INST] Solving System of Symmetric Equations I am to solve the system below. 
$$\begin{cases} (x^2+1)(y^2+1)=10 \\ (x+y)(xy-1)=3 \end{cases}$$
I have tried to factor the equations of the system but they do not factor. How can I approach the problem?
 [/INST] We have $$x^2+y^2+x^2y^2=9$$ or
$$(x+y)^2-2xy+x^2y^2=9$$  and since $$x+y=\frac{3}{xy-1},$$ we obtain:
$$\frac{9}{(xy-1)^2}-2xy+x^2y^2=9$$ or
$$\frac{9(2-xy)xy}{(xy-1)^2}-xy(2-xy)=0$$ or
$$xy(2-xy)(4-xy)(2+xy)=0.$$
Now, $xy=0$ gives $(-3,0)$ and $(0,-3)$;
$xy=2$ gives $x+y=3$ and $(2,1)$ and $(1,2)$;
$xy=4$ does not give real solutions 
and $xy=-2$ gives $x+y=-1$ and $(-2,1)$ and $(1,-2).$
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