Example 6 Solve the equation $\frac{13 x-x^{2}}{x+1}\left(x+\frac{13-x}{x+1}\right)=42$. (1998, Changchun City, Jilin Province Mathematics Competition (Grade 9))
Solve the original equation by transforming it as follows:
$$
\begin{array}{l}
\frac{13 x-x^{2}}{x+1}\left(\frac{x^{2}+13}{x+1}\right)=42 . \\
\text { Let } \frac{13 x-x^{2}}{x+1}=a+b, \\
\frac{x^{2}+13}{x+1}=a-b .
\end{array}
$$
Multiplying (1) and (2), and adding (1) and (2) respectively, we get
$$
a^{2}-b^{2}=42 \text {, }
$$
$$
2 a=\frac{13(x+1)}{x+1}=13 \text {. }
$$

Solving equations (3) and (4) simultaneously, we get \( a=\frac{13}{2}, b= \pm \frac{1}{2} \). Substituting into equation (2) gives \( \frac{x^{2}+13}{x+1}=7 \) or 6.
Solving these, we get \( x_{1}=1, x_{2}=6, x_{3}=3+\sqrt{2}, x_{4}=3-\sqrt{2} \). Upon verification, \( x_{1}, x_{2}, x_{3}, x_{4} \) are all roots of the original equation.