The given expression is:

$5\sqrt{2} + 4\sqrt{6} - 8\sqrt{2} + \sqrt{6}$

Let's simplify it by combining like terms. Notice that the terms with $\sqrt{2}$ can be grouped together, and the terms with $\sqrt{6}$ can also be grouped.

Step-by-step simplification:

1. Group the $\sqrt{2}$ terms: $5\sqrt{2} - 8\sqrt{2} = (5 - 8)\sqrt{2} = -3\sqrt{2}$

2. Group the $\sqrt{6}$ terms: $4\sqrt{6} + \sqrt{6} = (4 + 1)\sqrt{6} = 5\sqrt{6}$

Final simplified expression:

$-3\sqrt{2} + 5\sqrt{6}$

This is the simplified form of the given expression.

Would you like more details or have any questions?

Related questions:

1. How do you add or subtract terms with square roots?
2. Can square roots of different numbers be combined?
3. What is the general rule for simplifying square root expressions?
4. How can you rationalize expressions with square roots?
5. How does simplifying radical expressions relate to solving quadratic equations?

Tip:

Always group terms with the same square root when simplifying radical expressions.