\(\Delta=m^2+2m-3\ge0\Rightarrow\left[{}\begin{matrix}m\ge1\\m\le-3\end{matrix}\right.\)
\(\left\{{}\begin{matrix}x_1^2+5x_2^2=6\\x_1x_2=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x_1^2+5x_2^2=6\\x_2=\frac{1}{x_1}\end{matrix}\right.\)
\(\Rightarrow x_1^2+\frac{5}{x_1^2}=6\Leftrightarrow x_1^4-6x_1^2+5=0\Rightarrow\left[{}\begin{matrix}x_1^2=1\\x_1^2=5\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x_1=1\Rightarrow x_2=1\\x_1=-1\Rightarrow x_2=-1\\x_1=\sqrt{5}\Rightarrow x_2=\frac{1}{\sqrt{5}}\\x_1=-\sqrt{5}\Rightarrow x_2=-\frac{1}{\sqrt{5}}\end{matrix}\right.\) thay vào \(x_1+x_2=-\left(m+1\right)\Rightarrow m=-\left(x_1+x_2+1\right)\)
\(\Rightarrow\left[{}\begin{matrix}m=-3\\m=1\\m=-\frac{5+6\sqrt{5}}{5}\\m=\frac{-5+6\sqrt{5}}{5}\end{matrix}\right.\)