To solve the given expression, we need to simplify the numerator and the denominator separately and then divide them.

The given expression is:

\[ \frac{\left( \sqrt{3} + \sqrt{5} \right) \cdot \left( 3 + \sqrt{5} \right)}{\sqrt{10} + \sqrt{2}} \]

First, let's simplify the numerator:

\[ \left( \sqrt{3} + \sqrt{5} \right) \cdot \left( 3 + \sqrt{5} \right) \]

Using the distributive property (FOIL method):

\[ = \sqrt{3} \cdot 3 + \sqrt{3} \cdot \sqrt{5} + \sqrt{5} \cdot 3 + \sqrt{5} \cdot \sqrt{5} \]
\[ = 3\sqrt{3} + \sqrt{15} + 3\sqrt{5} + 5 \]

So, the numerator simplifies to:

\[ 3\sqrt{3} + \sqrt{15} + 3\sqrt{5} + 5 \]

Now, let's look at the denominator:

\[ \sqrt{10} + \sqrt{2} \]

So, the expression becomes:

\[ \frac{3\sqrt{3} + \sqrt{15} + 3\sqrt{5} + 5}{\sqrt{10} + \sqrt{2}} \]

This is the simplified form of the given expression. Further simplification might not be straightforward without additional context or constraints.