The given expression is:

\[
\frac{6x^4 - 18x^2}{2x^3 - 6x}
\]

Let's simplify it step by step.

### Step 1: Factor both the numerator and the denominator.

- Numerator: \(6x^4 - 18x^2\)
    - Factor out the common term \(6x^2\):
    \[
    6x^2(x^2 - 3)
    \]

- Denominator: \(2x^3 - 6x\)
    - Factor out the common term \(2x\):
    \[
    2x(x^2 - 3)
    \]

Now the expression becomes:

\[
\frac{6x^2(x^2 - 3)}{2x(x^2 - 3)}
\]

### Step 2: Simplify the fraction.

The \((x^2 - 3)\) term in both the numerator and denominator cancels out (assuming \(x^2 \neq 3\) to avoid division by zero):

\[
\frac{6x^2}{2x}
\]

### Step 3: Simplify the remaining expression.

\[
\frac{6x^2}{2x} = 3x
\]

Thus, the simplified form of the expression is:

\[
3x
\]

### Final answer:

\[
3x
\]

Would you like further details or have any questions?

---

Here are 5 related questions to expand your understanding:
1. What happens if \(x^2 - 3 = 0\)?
2. How would the expression change if the powers of \(x\) were different?
3. How do you handle cases where terms don't cancel out?
4. What if there were higher powers of \(x\) in both the numerator and denominator?
5. Can this be simplified further if \(x\) is a specific value?

**Tip**: Always check for common factors before simplifying algebraic expressions!

Learn how to simplify the rational expression 6x^4 - 18x^2 / 2x^3 - 6x with a detailed, step-by-step solution. This problem covers factoring, canceling common terms, and simplifying algebraic expressions. Suitable for students in Grades 9-10.

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