Example 4. Simplify: $\frac{1-a^{2}}{(1+a x)^{2}-(a+x)^{2}}$,
(Shanghai Mathematics Competition Question)
$$
\begin{array}{l}
=\frac{(1+a)(1-a)}{[1+(a+x)+a x][1-(a+x)+a x]} \\
=\frac{(1+a)(1-a)}{(1+a)(1+x)(1-a)(1-x)} \\
=\frac{1}{(1+x)} \frac{1}{(1-x)} \text {. } \\
\end{array}
$$

The above text translated into English, keeping the original text's line breaks and format, is as follows:

$$
\begin{array}{l}
=\frac{(1+a)(1-a)}{[1+(a+x)+a x][1-(a+x)+a x]} \\
=\frac{(1+a)(1-a)}{(1+a)(1+x)(1-a)(1-x)} \\
=\frac{1}{(1+x)} \frac{1}{(1-x)} \text {. } \\
\end{array}
$$