The problem asks us to simplify the expression $5\sqrt{3x} + 4\sqrt{48x} + 2\sqrt{75x}$, assuming that all variables represent nonnegative real numbers.

1. Problem Description

The problem asks us to simplify the expression $5\sqrt{3x} + 4\sqrt{48x} + 2\sqrt{75x}$, assuming that all variables represent nonnegative real numbers.

2. Solution Steps

First, simplify the radicals $\sqrt{48x}$ and $\sqrt{75x}$.

$\sqrt{48x} = \sqrt{16 \cdot 3x} = \sqrt{16} \cdot \sqrt{3x} = 4\sqrt{3x}$.

$\sqrt{75x} = \sqrt{25 \cdot 3x} = \sqrt{25} \cdot \sqrt{3x} = 5\sqrt{3x}$.

Now, substitute these simplified expressions back into the original expression:

$5\sqrt{3x} + 4\sqrt{48x} + 2\sqrt{75x} = 5\sqrt{3x} + 4(4\sqrt{3x}) + 2(5\sqrt{3x})$

$= 5\sqrt{3x} + 16\sqrt{3x} + 10\sqrt{3x}$.

Combine the terms with the same radical:

$5\sqrt{3x} + 16\sqrt{3x} + 10\sqrt{3x} = (5+16+10)\sqrt{3x} = 31\sqrt{3x}$.

3. Final Answer

$31\sqrt{3x}$