Problem 3. If $x y=6$ and $x^{2} y+x y^{2}+x+y=63$, then what is $x^{2}+y^{2}$?
Solution. From $x^{2} y+x y^{2}+x+y=63$ it follows that

$$
x y(x+y)+(x+y)=(x y+1)(x+y)=63 \Leftrightarrow 7(x+y)=63 \Leftrightarrow x+y=9
$$

Therefore,

$$
x^{2}+y^{2}=(x+y)^{2}-2 x y=81-12=69
$$