\(5x^2+5y^2+8xy-2x+2y+2=0\Leftrightarrow x^2+4x^2+y^2+4y^2+8xy-2x+2y+1+1=0\Leftrightarrow\left(x^2-2x+1\right)+\left(4x^2+8xy+4y^2\right)+\left(y^2+2y+1\right)=0\Leftrightarrow\left(x-1\right)^2+4\left(x+y\right)^2+\left(y+1\right)^2=0\)
Mà \(\left\{{}4\begin{matrix}\left(x-1\right)^2\ge0\\\left(x+y\right)^2\ge0\\\left(y+1\right)^2\ge0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\left(x-1\right)^2=0\\4\left(x+y\right)^2=0\\\left(y+1\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x-1=0\\x+y=0\\y+1=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=1\\x=-y\\y=-1\end{matrix}\right.\)
Với \(x=1;y=-1\) ta có:
\(M=\left(x+y\right)^{2016}+\left(x-2\right)^{2017}+\left(y+1\right)^{2018}=\left(1-1\right)^{2016}+\left(1-2\right)^{2017}+\left(-1+1\right)^{2018}=0+\left(-1\right)+0=-1\)
Vậy M = -1
