$$\begin{aligned} & A X=B \\ & 2 x-5 y=20 \\ & 3 x+m y=m \\ & \Rightarrow 3\left(\frac{20+5 y}{2}\right)+m y=m \end{aligned}$$

$$\begin{aligned} & \Rightarrow 30+\frac{15}{2} y+m y=m \\ & \Rightarrow y\left(\frac{15}{2}+m\right)=m-30 \\ & \Rightarrow y=\frac{m-30}{\frac{15}{2}+m}<0 \Rightarrow m \in\left(-\frac{15}{2}, 30\right) \end{aligned}$$

Similarly : $$3 x+m\left(\frac{2 x-20}{5}\right)=m$$

$$\begin{aligned} \Rightarrow & 3 x+\frac{2 m x}{5}-\frac{20 m}{5}=m \\ \Rightarrow & \frac{15 x+2 m x}{5}=5 m \Rightarrow x=\frac{25 m}{15+2 m} \\ & x<0 \Rightarrow \frac{25 m}{15+2 m}<0 \Rightarrow m \in\left(-\frac{15}{2}, 0\right) \\ \therefore \quad & m \in\left(-\frac{15}{2}, 0\right) \\ & a=-\frac{15}{2}, b=0 \\ & 8 \int_\limits{-\frac{15}{2}}^0(2 m+15) d m=450 \\ \end{aligned}$$