What is the following simplified product? Assume \( x \geq 0 \) \( 2 \sqrt{8 x^{3}}\left(3 \sqrt{10 x^{4}}-x \sqrt{5 x^{2}}\right) \)

Let's start by simplifying the expression step by step. 1. **Simplifying \( 2 \sqrt{8x^3} \):** \[ \sqrt{8x^3} = \sqrt{8} \sqrt{x^3} = 2\sqrt{2} \cdot x \sqrt{x} = 2\sqrt{2} x^{3/2} \] Thus, \[ 2 \sqrt{8x^3} = 2 \cdot 2\sqrt{2} x^{3/2} = 4\sqrt{2} x^{3/2} \] 2. **Simplifying \( 3 \sqrt{10x^4} - x \sqrt{5x^2} \):** - For \( 3 \sqrt{10x^4} \): \[ \sqrt{10x^4} = \sqrt{10} \cdot \sqrt{x^4} = \sqrt{10} \cdot x^2 \] Therefore, \[ 3 \sqrt{10x^4} = 3\sqrt{10} x^2 \] - For \( x \sqrt{5x^2} \): \[ x \sqrt{5x^2} = x \cdot \sqrt{5} \cdot x = x^2 \sqrt{5} \] Combining these gives: \[ 3\sqrt{10} x^2 - x^2 \sqrt{5} = x^2 (3\sqrt{10} - \sqrt{5}) \] 3. **Putting it all together:** Now substitute back into the expression: \[ 4\sqrt{2} x^{3/2} (3\sqrt{10} - \sqrt{5}) x^2 \] Simplifying further: \[ = 4\sqrt{2} (3\sqrt{10} - \sqrt{5}) x^{3/2 + 2} = 4\sqrt{2} (3\sqrt{10} - \sqrt{5}) x^{7/2} \] Thus, the simplified product is: \[ \boxed{4\sqrt{2} (3\sqrt{10} - \sqrt{5}) x^{7/2}} \]