The problem asks us to simplify the expression $(2\sqrt{5} - \frac{\sqrt{5}}{3}) + (5\sqrt{2} + 5\sqrt{2})$.

1. Problem Description

The problem asks us to simplify the expression $(2\sqrt{5} - \frac{\sqrt{5}}{3}) + (5\sqrt{2} + 5\sqrt{2})$.

2. Solution Steps

First, let's simplify the first parenthesis:

$2\sqrt{5} - \frac{\sqrt{5}}{3} = \frac{6\sqrt{5}}{3} - \frac{\sqrt{5}}{3} = \frac{6\sqrt{5} - \sqrt{5}}{3} = \frac{5\sqrt{5}}{3}$

Next, let's simplify the second parenthesis:

$5\sqrt{2} + 5\sqrt{2} = 10\sqrt{2}$

Now, we add the two simplified terms:

$\frac{5\sqrt{5}}{3} + 10\sqrt{2}$

3. Final Answer

The final answer is $\frac{5\sqrt{5}}{3} + 10\sqrt{2}$.