Sure! Let's break down the given expression [tex]\(\frac{4 x^3 - \frac{2}{x}}{3 x^2 - \sqrt{x}}\)[/tex] step by step.
First, we analyze the numerator and the denominator separately.
### Numerator:
The numerator of our given expression is [tex]\(4x^3 - \frac{2}{x}\)[/tex].
### Denominator:
The denominator of our expression is [tex]\(3x^2 - \sqrt{x}\)[/tex].
### Combined Expression:
Now, we need to form the fraction using the numerator and denominator:
[tex]\[
\frac{4x^3 - \frac{2}{x}}{3x^2 - \sqrt{x}}
\][/tex]
### Simplifying the Expression:
Next, we simplify the expression. The simplified form of our fraction is as follows:
[tex]\[
\frac{4x^3 - \frac{2}{x}}{3x^2 - \sqrt{x}} = \frac{2(2x^4 - 1)}{-x^{\frac{3}{2}} + 3x^3}
\][/tex]
### Final Simplified Form:
The final simplified form of the given expression is [tex]\(\frac{2(2x^4 - 1)}{-x^{\frac{3}{2}} + 3x^3}\)[/tex].
So, in summary, the expression [tex]\(\frac{4x^3 - \frac{2}{x}}{3x^2 - \sqrt{x}}\)[/tex] simplifies to [tex]\(\frac{2(2x^4 - 1)}{-x^{\frac{3}{2}} + 3x^3}\)[/tex].
