The reflection

Since $B(m,n)$ is the reflection of the point $A(10,9)$ across the line $l$ , $4x-3y=12$ :

1.) The midpoint of line segment $AB$ lies on line $l$

2.) Line joining $A$ and $B$ is perpendicular to line $l$

The midpoint of line segment $AB$ satisfies the equation of line $l$ $M=\left(\frac{10+m}{2},\,\frac{9+n}{2}\right)$ $\begin{aligned} &\implies 4\left(\frac{10+m}{2}\right)-3\left(\frac{9+n}{2}\right)+12=0 \\ &\implies 4m-3n+37=0 \end{aligned}$ The product of the slope of line $AB$ and the slope of $l$ is $-1$ $\frac{n-9}{m-10}\cdot \frac{4}{3}=-1$ $\implies 3m+4n-66=0\;\;\;\;\;\;\;\;$ Solving the system of equations, $\begin{cases} 4m-3n+37=0\\ 3m+4n-66=0\end{cases}$ we get, $m=2$ and $n=15$ . Therefore, $n+(n\times m)-m=\boxed{43}$