Example 4 Let $m$ be an integer, and $4<m<40$, and 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.
Analysis: Considering the discriminant $\Delta=4(2 m+1)$, since it is a linear expression in terms of $m$, the methods used in Example 1 and Example 2 are not applicable.
Given $4<m<40$, we know that
$9<2 m+1<81$.
To make the discriminant a perfect square, $2 m+1$ can only be $25$ or $2 m+1=49$.

When $2 m+1=25$, $m=12$, the roots of the equation are 16 and 26;

When $2 m+1=49$, $m=24$, the roots of the equation are 38 and 52.

Note: When the discriminant is not a quadratic expression, the parameter range can be determined by combining the given conditions through discussion, and then solve the problem; when the discriminant is more complex, other methods should be used, refer to Example 5.