Certainly! Let's simplify the given expression step by step. We have:
[tex]\[
\sqrt{5x} \left( \sqrt{8x^2} - 2\sqrt{x} \right)
\][/tex]
First, let’s handle the terms inside the parentheses separately:
1. Simplify [tex]\(\sqrt{8x^2}\)[/tex]:
[tex]\[
\sqrt{8x^2} = \sqrt{4 \cdot 2 \cdot x^2} = \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{x^2} = 2\sqrt{2} \cdot x
\][/tex]
2. Simplify [tex]\(\sqrt{x}\)[/tex]:
This is already in its simplest form as [tex]\(\sqrt{x}\)[/tex].
So now the expression inside the parentheses becomes:
[tex]\[
2\sqrt{2} \cdot x - 2\sqrt{x}
\][/tex]
Next, we distribute [tex]\(\sqrt{5x}\)[/tex] across the terms inside the parentheses:
[tex]\[
\sqrt{5x} \cdot \left( 2\sqrt{2} \cdot x - 2\sqrt{x} \right)
\][/tex]
We do this term-by-term:
1. Simplify [tex]\(\sqrt{5x} \cdot 2\sqrt{2} \cdot x\)[/tex]:
[tex]\[
\sqrt{5x} \cdot 2\sqrt{2} \cdot x = 2 \sqrt{10x^2} \cdot x = 2 \sqrt{10} \cdot |x| \cdot x
\][/tex]
Since [tex]\(x \geq 0\)[/tex], [tex]\(|x| = x\)[/tex]. So, this simplifies to:
[tex]\[
2 \cdot \sqrt{10} \cdot x \cdot x = 2\sqrt{10} \cdot x^2
\][/tex]
2. Simplify [tex]\(\sqrt{5x} \cdot -2\sqrt{x}\)[/tex]:
[tex]\[
\sqrt{5x} \cdot -2\sqrt{x} = -2 \cdot \sqrt{5x \cdot x} = -2 \cdot \sqrt{5x^2} = -2 \cdot \sqrt{5} \cdot |x|
\][/tex]
Again, since [tex]\(x \geq 0\)[/tex], [tex]\(|x| = x\)[/tex]. So, this simplifies to:
[tex]\[
-2 \cdot \sqrt{5} \cdot x
\][/tex]
Putting these pieces together, the entire expression simplifies to:
[tex]\[
2\sqrt{10} \cdot x^2 - 2\sqrt{5} \cdot x
\][/tex]
Therefore, the simplified product is:
[tex]\[
2\sqrt{10} \cdot x^{3/2} - 2\sqrt{5} \cdot x = 2\sqrt{10} \cdot x \cdot x - 2\sqrt{5} x
\][/tex]
So the correct choice is:
[tex]\[
2 x \sqrt{10 x}-2 x \sqrt{5}
\][/tex]
So the correct answer from the given options is:
[tex]\[
2x\sqrt{10x} - 2x\sqrt{5}
\][/tex]
