To simplify the expression [tex]\(\sqrt{50 x^3}-\sqrt{25 x^3}+5 \sqrt{x^3}-\sqrt{2 x^3}\)[/tex], let us handle each term separately and then combine them.
1. Simplify [tex]\(\sqrt{50 x^3}\)[/tex]:
[tex]\[
\sqrt{50 x^3} = \sqrt{50} \cdot \sqrt{x^3}
\][/tex]
Knowing that [tex]\(\sqrt{x^3} = x^{3/2}\)[/tex], we have:
[tex]\[
\sqrt{50} \cdot x^{3/2} = \sqrt{25 \cdot 2} \cdot x^{3/2} = 5\sqrt{2} \cdot x^{3/2}
\][/tex]
2. Simplify [tex]\(\sqrt{25 x^3}\)[/tex]:
[tex]\[
\sqrt{25 x^3} = \sqrt{25} \cdot \sqrt{x^3}
\][/tex]
[tex]\[
\sqrt{25} \cdot x^{3/2} = 5 \cdot x^{3/2}
\][/tex]
3. Simplify [tex]\(5 \sqrt{x^3}\)[/tex]:
[tex]\[
5 \sqrt{x^3} = 5 \cdot x^{3/2}
\][/tex]
4. Simplify [tex]\(\sqrt{2 x^3}\)[/tex]:
[tex]\[
\sqrt{2 x^3} = \sqrt{2} \cdot \sqrt{x^3}
\][/tex]
[tex]\[
\sqrt{2} \cdot x^{3/2}
\][/tex]
Now, let's combine the simplified terms:
[tex]\[
5\sqrt{2} \cdot x^{3/2} - 5 \cdot x^{3/2} + 5 \cdot x^{3/2} - \sqrt{2} \cdot x^{3/2}
\][/tex]
Group like terms (terms with [tex]\( x^{3/2} \)[/tex]):
[tex]\[
(5\sqrt{2} - 5 + 5 - \sqrt{2}) x^{3/2}
\][/tex]
Simplify the coefficients:
[tex]\[
(5\sqrt{2} - \sqrt{2}) x^{3/2}
\][/tex]
[tex]\[
(4\sqrt{2}) x^{3/2}
\][/tex]
[tex]\[
4 \sqrt{2} \cdot x^{3/2}
\][/tex]
Rewrite [tex]\( x^{3/2} \)[/tex] as [tex]\( x \cdot \sqrt{x} \)[/tex]:
[tex]\[
4 \sqrt{2} \cdot x \cdot \sqrt{x}
\][/tex]
[tex]\[
4x \sqrt{2x}
\][/tex]
Therefore, the expression simplifies to [tex]\( 4x \sqrt{2x} \)[/tex], corresponding to choice D.
So the answer is:
[tex]\[
\boxed{4 x \sqrt{2 x}}
\][/tex]
