Example 5. Simplify $\frac{\sqrt{5}+3 \sqrt{3}+4 \sqrt{2}}{(\sqrt{5}+\sqrt{2})(\sqrt{2}+\sqrt{3})}$. (1994, Tianjin Mathematical Competition)
Analyzing, if we rationalize the denominator in the conventional way, it would be very complicated. Observing the characteristics of the numerator, we get
the original expression
$$
\begin{array}{l}
=\frac{(\sqrt{5}+\sqrt{2})+3(\sqrt{2}+\sqrt{3})}{(\sqrt{5}+\sqrt{2})(\sqrt{2}+\sqrt{3})} \\
=\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{3}{\sqrt{5}+\sqrt{2}} \\
=\sqrt{3}+\sqrt{5}-2 \sqrt{2} .
\end{array}
$$