Example 6 If the solution set of the inequality $(m-3) x^{2}-2 m x-8>0$ with respect to $x$ is an open interval, and the length $l$ of the interval satisfies $l \in[1,2]$, find the range of real number $m$ (Note: The length $l$ of an open interval $(a, b)$ is $l=b-a$).
Solve: According to the problem, we get $m-30 \Leftrightarrow\left(x-x_{1}\right)(x-$ $\left.x_{2}\right)0$, which leads to $m>-4+2 \sqrt{10}$ or $m-15,-4+2 \sqrt{10}<\frac{7}{3}$, so the range of $m$ is $(-\infty,-15] \cup$ $\left[\frac{7}{3}, \frac{33}{14}\right]$.