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?

Here is a solution using vieta's formula. 
Let $s=x+y$ and $p=xy$. Then $s+p=44$ and $sp=448$. Therefore $s$ and $p$ are the roots of the quadratic $z^2-44z+448=0$. Thus it follows that $s=16$ and $p=28$. Hence $$x^2+y^2=(x+y)^2-2xy=s^2-2p=256-56=\boxed{200}.$$
But note that if $s=28$ and $p=16$, then 
$$x^2+y^2=(x+y)^2-2xy=784-32=\boxed{752}.$$
