Example 7 Let $\alpha$ be an integer. If there exist integers $b$ and $c$ such that $(x+\alpha)(x-15)-25=(x+b)(x+c)$ holds, find the possible values of $\alpha$.


Analysis: This problem can be transformed into: For what value of $\alpha$ does the equation $(x+\alpha)(x-15)-25=0$ have integer roots.
The equation can be rewritten as
$x^{2}-(15-\alpha) x-15 \alpha-25=0$.
Considering it as a linear equation in $\alpha$, we rearrange to get
$$
\alpha(x-15)=-x^{2}+15 x+25 \text {. }
$$

It is clear that $x \neq 15$,
$$
\therefore \alpha=\frac{-x^{2}+15 x+25}{x-15}=-x+\frac{25}{x-15} \text {. }
$$

Since $\alpha$ and $x$ are both integers, the discussion is as follows:
\begin{tabular}{|c|r|r|r|r|}
\hline$x-15$ & -25 & -5 & 5 & 25 \\
\hline$x$ & -10 & 10 & 20 & 40 \\
\hline$\alpha$ & 9 & -15 & -15 & -39 \\
\hline
\end{tabular}
The values of $\alpha$ are: $9, -15, -39$, and upon verification, they satisfy the problem's conditions.
Note: This solution method is the parameter separation method, which is suitable for cases where the roots of the equation are all integers and the parameter is relatively easy to separate. In this problem, the equation is transformed into $\alpha=f(x)$, and then the solution is found using the properties of the function, which is a widely used method.

The above text only introduces the commonly used methods for solving integer root problems. The basic basis of these methods is: the necessary condition for the equation to have integer roots. The basic methods are: (1) discriminant discussion method (mainly discussing the parameter range determined by the discriminant, and finding the parameter when the discriminant is a perfect square): (2) Vieta's theorem method; (3) combination of discriminant and Vieta's theorem method; (4) parameter separation method (separating the parameter and using the condition that the roots are integers for discussion).

It should be noted that the solution method for each problem is not unique. The one provided here is just a more concise one. When solving problems, students should choose methods based on the problem and strive for innovation to appreciate the beauty of mathematics.