\(\left(x+y\right)^2=\left(x-2\right)\left(y+2\right)\)
\(\Leftrightarrow x^2+y^2+2xy=xy+2x-2y-4\)
\(\Leftrightarrow x^2+y^2+xy-2x+2y+4=0\)
\(\Leftrightarrow2x^2+2y^2+2xy-4x+4y+8=0\)
\(\Leftrightarrow\left(x^2+2xy+y^2\right)+\left(x^2-4x+4\right)+\left(y^2+4y+4\right)=0\)
\(\Leftrightarrow\left(x+y\right)^2+\left(x-2\right)^2+\left(y+2\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+y=0\\x-2=0\\y+2=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=-2\end{matrix}\right.\)
\(\Rightarrow A^2=2^2+\left(-2\right)^2=8\)
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