Sure, let's simplify the given rational function step-by-step:
[tex]\[
\frac{6 x^3 - 8 x^2 - 6 x + 3}{2 x^2}
\][/tex]
### Step 1: Write Down the Expression
We start with the given expression:
[tex]\[
\frac{6 x^3 - 8 x^2 - 6 x + 3}{2 x^2}
\][/tex]
### Step 2: Separate Each Term in the Numerator by the Denominator
We can break down the expression by separating each term in the numerator with the denominator:
[tex]\[
\frac{6 x^3}{2 x^2} - \frac{8 x^2}{2 x^2} - \frac{6 x}{2 x^2} + \frac{3}{2 x^2}
\][/tex]
### Step 3: Simplify Each Fraction Individually
1. Simplify [tex]\(\frac{6 x^3}{2 x^2}\)[/tex]:
[tex]\[
\frac{6 x^3}{2 x^2} = 3 x
\][/tex]
2. Simplify [tex]\(\frac{8 x^2}{2 x^2}\)[/tex]:
[tex]\[
\frac{8 x^2}{2 x^2} = 4
\][/tex]
3. Simplify [tex]\(\frac{6 x}{2 x^2}\)[/tex]:
[tex]\[
\frac{6 x}{2 x^2} = \frac{6}{2 x} = \frac{3}{x}
\][/tex]
4. Simplify [tex]\(\frac{3}{2 x^2}\)[/tex]:
[tex]\[
\frac{3}{2 x^2} \text{ (This term cannot be simplified further in terms of } x \text{)}
\][/tex]
### Step 4: Combine the Simplified Terms
Now, we combine the simplified terms:
[tex]\[
3 x - 4 - \frac{3}{x} + \frac{3}{2 x^2}
\][/tex]
### Final Result
The simplified form of the given rational expression is:
[tex]\[
3 x - 4 - \frac{3}{x} + \frac{3}{2 x^2}
\][/tex]
This is the simplified expression of [tex]\(\frac{6 x^3 - 8 x^2 - 6 x + 3}{2 x^2}\)[/tex].
