Example 3 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 $l: x-y+1=0$.
Solution: Given $l: x=y-1, y=x+1$, substituting into equation $E$ yields $2(y-1)^{2}+4(y-1)(x+1)+5(x+1)^{2}-$ $22=0$, simplifying to $5 x^{2}+4 x y+2 y^{2}+6 x-19=0$, which is the desired result.