<s>[INST] Solving an equation by telling the value of $x^2+y^2$. I have a problem solving an equation. The equation is:
$xy+x+y=44$ and $x^2y+xy^2=448$
and we have to tell the value of $x^2+y^2$
First I tried solving this by doing the following:
$xy+x+y=44~\to~x+y=44-xy~\to~x^2+2xy+y^2=44^2-88xy+x^2y^2~\Rightarrow$
$\Rightarrow~x^2+y^2=44^2-90xy+x^2y^2$
But from here I didn't know what to do. Could you help me in solving this equation?
 [/INST] Let $A = xy$ and $B = x + y$. Then we have
$$A + B = 44$$
$$AB = 448$$
By Viete's formula, $A$ and $B$ are the roots of
$$r^2 - 44r + 448 = 0$$
$$(r - 16)(r - 28) = 0$$
such that
$$(xy, x + y) = (16, 28),~(28, 16)$$ 
from which it is trivial do deduce $x^2 + y^2$:
$$\begin{align}x^2 + y^2 &= (x + y)^2 - 2xy
\\&= 28^2 - 2(16), 16^2 - 2(28)\\
&=752, 200\end{align}$$
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