Question

Answer 1

\(M=\left(x^2+2xy+y^2\right)+2\left(x+y\right)+1+\left(y^2-4y+4\right)-2021\\ M=\left[\left(x+y\right)^2+2\left(x+y\right)+1\right]+\left(y-2\right)^2-2021\\ M=\left(x+y+1\right)^2+\left(y-2\right)^2-2021\ge-2021\\ M_{min}=-2021\Leftrightarrow\left\{{}\begin{matrix}x=-y-1\\y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-3\\y=2\end{matrix}\right.\)