\(\Leftrightarrow\left\{{}\begin{matrix}4x+2y=10m-2\\x-2y=2\end{matrix}\right.\) \(\Rightarrow5x=10m\Rightarrow x=2m\)
\(\Rightarrow y=5m-1-2x=m-1\)
\(x^2-2y^2=-2\Leftrightarrow\left(2m\right)^2-2\left(m-1\right)^2=-2\)
\(\Leftrightarrow4m^2-2m^2+4m-2+2=0\)
\(\Leftrightarrow2m^2+4m=0\Leftrightarrow2m\left(m+2\right)=0\Rightarrow\left[{}\begin{matrix}m=0\\m=-2\end{matrix}\right.\)