#### Answer

$\dfrac{3-\sqrt{5}+3\sqrt{2}-\sqrt{10}}{4}$

#### Work Step by Step

Multiplying by the conjugate of the denominator, the rationalized-denominator form of the given expression, $
\dfrac{1+\sqrt{2}}{3+\sqrt{5}}
,$ is
\begin{array}{l}\require{cancel}
\dfrac{1+\sqrt{2}}{3+\sqrt{5}}\cdot\dfrac{3-\sqrt{5}}{3-\sqrt{5}}
\\\\=
\dfrac{1(3)+1(-\sqrt{5})+\sqrt{2}(3)+\sqrt{2}(-\sqrt{5})}{3^2-(\sqrt{5})^2}
\\\\=
\dfrac{3-\sqrt{5}+3\sqrt{2}-\sqrt{2(5)}}{9-5}
\\\\=
\dfrac{3-\sqrt{5}+3\sqrt{2}-\sqrt{10}}{4}
.\end{array}