Example 4 Given $u+v=96$, and the quadratic equation $x^{2} + u x + v = 0$ has integer roots, then its largest root is $\qquad$
(1996, Anhui Province Partial Areas Junior High School Mathematics League)
Solution: Let the two integer roots of the equation be $x_{1}$ and $x_{2}$, then we have
$$
\begin{array}{l}
x_{1}+x_{2}=-u, x x_{2}=v . \\
\because u+v=96, \\
\therefore x_{1} x_{2}-\left(x_{1}+x_{2}\right)=v+u=96 .
\end{array}
$$

Thus, $x_{1} x_{2}-x_{1}-x_{2}+1=97$,
which means $\left(x_{1}-1\right)\left(x_{2}-1\right)=97$.
Since 97 is a prime number, then
$x_{1}-1=97$ or $x_{2}-1=97$.
$\therefore x_{1}=98$ or $x_{2}=98$.
Therefore, the largest root of the equation is 98.