Example 4 Find the equation of the curve $E^{\prime}$ symmetric to the curve $E: 2 x^{2}+4 x y+5 y^{2}-22=0$ with respect to the line $l: x-2 y+1=0$.
Solution: Let $M^{\prime}(x, y)$ be any point on the curve $E^{\prime}$, and the point symmetric to $M^{\prime}(x, y)$ with respect to the line $l$ be $M\left(x_{1}, y_{1}\right)$. By formula (*), we have
$$
x_{1}=\frac{3 x+4 y-2}{5}, y_{1}=-\frac{-3 y+4 x+4}{5} \text {, }
$$

Substituting into the equation of $E$, we get
$$
\begin{array}{l}
2\left(\frac{3 x+4 y-2}{5}\right)^{2} \\
+4\left(\frac{3 x+4 y-2}{5}\right)\left(\frac{-3 y+4 x+4}{5}\right) \\
+5\left(\frac{-3 y+4 x+4}{5}\right)^{2}-22=0,
\end{array}
$$

Simplifying, we get $146 x^{2}-44 x y+29 y^{2}+152 x-64 y-$ $494=0$, which is the desired equation.