You provided the expression:

\[
\frac{x^2 - 4x + 6x^3}{2x}
\]

To simplify this expression, let's go step by step:

### Step 1: Split the terms
We can split the fraction into separate terms, dividing each term in the numerator by \( 2x \):

\[
\frac{x^2}{2x} - \frac{4x}{2x} + \frac{6x^3}{2x}
\]

### Step 2: Simplify each term
- For the first term: \( \frac{x^2}{2x} = \frac{1}{2}x \)
- For the second term: \( \frac{4x}{2x} = 2 \)
- For the third term: \( \frac{6x^3}{2x} = 3x^2 \)

### Step 3: Combine the simplified terms
Putting everything together, the simplified expression is:

\[
\frac{1}{2}x - 2 + 3x^2
\]

Thus, the simplified expression is:

\[
3x^2 + \frac{1}{2}x - 2
\]

Would you like more details on any of the steps?

### Related questions:
1. How would you simplify \( \frac{3x^2 - 5x + 2x^3}{x} \)?
2. Can you solve the equation \( \frac{x^2 - 4x + 6x^3}{2x} = 0 \)?
3. What happens if you multiply \( \frac{x^2 - 4x + 6x^3}{2x} \) by \( x \)?
4. How do you simplify \( \frac{5x^3 + 2x^2 - x}{x} \)?
5. How do you factor the expression \( 3x^2 + \frac{1}{2}x - 2 \)?

### Tip:
When simplifying rational expressions, always check for common factors in both the numerator and denominator that you can cancel out.

Learn how to simplify the rational expression x^2 - 4x + 6x^3 / 2x with a step-by-step solution. This problem involves key algebraic concepts like polynomial division and rational expressions, suitable for students in Grades 9-10.

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