If x and y are positive integers such that $xy+x+y=71,x^{2}y+x{y}^2=880$, then $x^2+y^2$ is equal to

$\because xy+x+y=71\Rightarrow xy+(x+y)=71$

and $x^{2}y+x{y}^2=880\Rightarrow xy(x+y)=880$

$\Rightarrow xy$ and $\left(x+y\right)$ are  the roots of the quadratic equation.

$\begin{array}{l}{t}^2-71t+880=0\\ \Rightarrow (t-55)(t-16)=0\\ \Rightarrow t=55,16\end{array}$

$\therefore x+y=16$ and $xy =55$

So, $x^2+y^2=(x+y{)}^2-2xy=(16{)}^2-110=146$