\(P=x^2-4xy+5y^2+10x-22y+2042\)
\(=\left[x^2-2x\left(2y-5\right)+\left(2y-5\right)^2\right]+\left(y^2-2y+1\right)+2016\)
\(=\left(x-2y+5\right)^2+\left(y-1\right)^2+2016\ge2016\)
\(minP=2016\Leftrightarrow\) \(\left\{{}\begin{matrix}x=-3\\y=1\end{matrix}\right.\)
\(P=\left(x^2-4xy+4y^2\right)+10\left(x-2y\right)+25+y^2-2y+1+2016\\ P=\left[\left(x-2y\right)^2+10\left(x-2y\right)+25\right]+\left(y-1\right)^2+2016\\ P=\left(x-2y+5\right)^2+\left(y-1\right)^2+2016\ge2016\\ P_{min}=2016\Leftrightarrow\left\{{}\begin{matrix}x-2y+5=0\\y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-3\\y=1\end{matrix}\right.\)
