For example. Let $m$ be an integer, for all $4<m<40$, the equation $x^{2}-2(2 m-3) x+4 m^{2}-14 m+8=0$ has two integer roots. Find the value of $m$ and the roots of the equation. (1984, Wuhan Preparatory Autumn Competition)
Solving, we get $\Delta=4(2 m+1)$. Given that $\Lambda=4(2 m +1)$ is a perfect square and $4<m<40$, it can be deduced that $m=12$ or 24, and the corresponding roots are $26,16$ and $52, 38$ (all integer roots). Therefore, the required values of $m$ are 12,24.
Note: If the discriminant is not a perfect square and its structure