Consider the matrices : $$A=\left[\begin{array}{cc}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$$ and $$X=\left[\begin{array}{l}x \\ y\end{array}\right]$$. Let the set of all $m$, for which the system of equations $A X=B$ has a negative solution (i.e., $x<0$ and $y<0$), be the interval $(a, b)$. Then $8 \int_\limits a^b|A| d m$ is equal to _________.

<p>$$\begin{aligned} & A X=B \\ & 2 x-5 y=20 \\ & 3 x+m y=m \\ & \Rightarrow 3\left(\frac{20+5 y}{2}\right)+m y=m \end{aligned}$$</p> <p>$$\begin{aligned} & \Rightarrow 30+\frac{15}{2} y+m y=m \\ & \Rightarrow y\left(\frac{15}{2}+m\right)=m-30 \\ & \Rightarrow y=\frac{m-30}{\frac{15}{2}+m}<0 \Rightarrow m \in\left(-\frac{15}{2}, 30\right) \end{aligned}$$</p> <p>Similarly : $3 x+m\left(\frac{2 x-20}{5}\right)=m$</p> <p>$$\begin{aligned} \Rightarrow & 3 x+\frac{2 m x}{5}-\frac{20 m}{5}=m \\ \Rightarrow & \frac{15 x+2 m x}{5}=5 m \Rightarrow x=\frac{25 m}{15+2 m} \\ & x<0 \Rightarrow \frac{25 m}{15+2 m}<0 \Rightarrow m \in\left(-\frac{15}{2}, 0\right) \\ \therefore \quad & m \in\left(-\frac{15}{2}, 0\right) \\ & a=-\frac{15}{2}, b=0 \\ & 8 \int_\limits{-\frac{15}{2}}^0(2 m+15) d m=450 \\ \end{aligned}$$</p>