Example 8. Solve the equation $\sqrt{\frac{x^{2}-2 x+3}{2 x^{2}+4 x+8}}+\sqrt{\frac{x^{2}+2 x+4}{x^{2}-2 x+3}}$
$$
=\frac{2+\sqrt{2}}{2} \text {. }
$$
Solve the original equation $\stackrel{\text { transformation }}{\Longleftrightarrow} \frac{1}{\sqrt{2}} \cdot \sqrt{\frac{x^{2}-2 x+8}{x^{2}+2 x+4}}+$ $\sqrt{\frac{x^{2}+2 x+4}{x^{2}-2 x+3}}=\frac{1}{\sqrt{2}}+1 \stackrel{\text { Proposition 1 }}{\Longleftrightarrow} \sqrt{\frac{x^{2}-2 x+3}{x^{2}+2 x+4}}$ $=1$ or $\sqrt{\frac{x^{2}-2 x+3}{x^{2}+2 x+4}}=\sqrt{2}$. Solving it yields $x_{1}=-\frac{1}{4}$, $x_{2}=-1, x_{3}=-5$ as the roots of the original equation.