\(A=2x^2+4y^2+4xy+10x+12y+18\)
\(A=x^2+4xy+4y^2+6x+12y+9+x^2+4x+4+5\)
\(A=\left(x+2y\right)^2+2.3\left(x+2y\right)+9+\left(x+2\right)^2+5\)
\(A=\left(x+2y+3\right)^2+\left(x+2\right)^2+5\)
Do : \(\left\{{}\begin{matrix}\left(x+2y+3\right)^2\ge0\forall x\\\left(x+2\right)^2\ge0\forall x\end{matrix}\right.\) \(\Leftrightarrow\left(x+2y+3\right)^2+\left(x+2\right)^2+5\ge5\)
\("="\Leftrightarrow\left\{{}\begin{matrix}x+2y+3=0\\x+2=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=-\dfrac{1}{2}\\x=-2\end{matrix}\right.\)
\(\Rightarrow A_{Min}=5\Leftrightarrow\left\{{}\begin{matrix}x=-2\\y=-\dfrac{1}{2}\end{matrix}\right.\)
