\(\left\{{}\begin{matrix}x-my=2-4m\\m^2x+my=3m^2+m\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x-my=2-4m\\\left(m^2+1\right)x=3m^2-3m+2\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x=\frac{3m^2-3m+2}{m^2+1}=3-\frac{3m+1}{m^2+1}\\y=\frac{4m^2+m+1}{m^2+1}=4-\frac{3-m}{m^2+1}\end{matrix}\right.\)
\(L=\left(3-\frac{3m+1}{m^2+1}\right)^2+\left(4-\frac{3-m}{m^2+1}\right)^2-6+\frac{6m+2}{m^2+1}\)
\(=19-\frac{4m+6}{m^2+1}\)
\(L_{max}\) khi \(k=\frac{4m+6}{m^2+1}\) đạt min
\(k=\frac{4m+6}{m^2+1}=km^2-4m+k-6=0\)
\(\Delta'=4-k\left(k-6\right)\ge0\)
\(\Leftrightarrow-k^2+6k+4\ge0\Rightarrow3-\sqrt{13}\le k\le3+\sqrt{13}\)
\(\Rightarrow L\le19-\left(3-\sqrt{13}\right)=16+\sqrt{13}\)