\(P=x^2+2y^2+2xy-4x-10y+21=\left[x^2+2x\left(y-2\right)+\left(y-2\right)^2\right]+\left(y^2-6y+9\right)+8=\left(x+y-2\right)^2+\left(y-3\right)^2+8\ge8\)
\(minP=8\Leftrightarrow\)\(\left\{{}\begin{matrix}x=-1\\y=3\end{matrix}\right.\)
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