$\left\{\begin{array}{l} 2x+y=5\\ -x+2y=10m+5\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} 2x+y=5\\ -2x+4y=20m+10\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} 2x+y=5\\ -2x+4y+2x+y=20m+10+5\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} 2x+y=5\\ 5y=20m+15\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} x=\dfrac{5-y}{2}\\ y=4m+3\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} x=-2m-1\\ y=4m+3\end{array} \right.\\ xy=(-2m-1)(4m+3)\\ =-8m^2-10m-3\\ =-\left(\left(2\sqrt{2}m\right)^2+2.2\sqrt{2}.\dfrac{5\sqrt{2}}{4}+\dfrac{25}{8}\right)+\dfrac{1}{8}\\ =-\left(2\sqrt{2}m+\dfrac{5\sqrt{2}}{4}\right)^2+\dfrac{1}{8} \le \dfrac{1}{8}$
Dấu "=" xảy ra $\Leftrightarrow 2\sqrt{2}m+\dfrac{5\sqrt{2}}{4}=0 \Leftrightarrow m=-\dfrac{5}{8}$