\(x^2+y^2+1=xy-x-y\Leftrightarrow2x^2+2y^2+2=2xy-2x-2y\)
\(\Leftrightarrow\left(x^2-2xy+y^2\right)+\left(x^2+2x+1\right)+\left(y^2+2y+1\right)=0\)
\(\Leftrightarrow\left(x-y\right)^2+\left(x+1\right)^2+\left(y+1\right)^2=0\Leftrightarrow x=y=-1\)
\(A=\frac{1}{xy}+2\left(x+y\right)=\frac{1}{\left(-1\right)\left(-1\right)}+2\left[\left(-1\right)+\left(-1\right)\right]=\frac{-7}{2}\)