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

The expression is:

\[ \frac{\sqrt{2} + \sqrt{3} + \sqrt{6} + \sqrt{8} + \sqrt{4}}{\sqrt{2} + \sqrt{3} + \sqrt{4}} \]

First, let's simplify the terms inside the square roots where possible:

- \(\sqrt{2}\) remains \(\sqrt{2}\)
- \(\sqrt{3}\) remains \(\sqrt{3}\)
- \(\sqrt{6}\) remains \(\sqrt{6}\)
- \(\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}\)
- \(\sqrt{4} = 2\)

Now substitute these simplified terms back into the expression:

Numerator:
\[ \sqrt{2} + \sqrt{3} + \sqrt{6} + 2\sqrt{2} + 2 \]

Combine like terms:
\[ (\sqrt{2} + 2\sqrt{2}) + \sqrt{3} + \sqrt{6} + 2 \]
\[ 3\sqrt{2} + \sqrt{3} + \sqrt{6} + 2 \]

Denominator:
\[ \sqrt{2} + \sqrt{3} + \sqrt{4} \]
\[ \sqrt{2} + \sqrt{3} + 2 \]

So the expression becomes:
\[ \frac{3\sqrt{2} + \sqrt{3} + \sqrt{6} + 2}{\sqrt{2} + \sqrt{3} + 2} \]

This is the simplified form of the given expression.