The given expression is:

\[
\frac{2x^2 - 4x}{2x^3 - 8x}
\]

### Step-by-Step Simplification:

1. **Factor both the numerator and the denominator:**

   - Numerator: \( 2x^2 - 4x \)
     \[
     = 2x(x - 2)
     \]

   - Denominator: \( 2x^3 - 8x \)
     \[
     = 2x(x^2 - 4) = 2x(x - 2)(x + 2)
     \]

2. **Rewrite the expression using these factored forms:**
   \[
   \frac{2x(x - 2)}{2x(x - 2)(x + 2)}
   \]

3. **Cancel out common factors**:
   The common factors \(2x(x - 2)\) appear in both the numerator and denominator, so they cancel out (as long as \(x \neq 0\) and \(x \neq 2\)):

   \[
   \frac{1}{x + 2}
   \]

### Final Answer:
\[
\frac{2x^2 - 4x}{2x^3 - 8x} = \frac{1}{x + 2} \quad \text{(for } x \neq 0, \, x \neq 2 \text{)}
\]

Let me know if you'd like more details!

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### Related Questions:
1. How do you factor polynomials like \(2x^2 - 4x\)?
2. What are the conditions under which we can cancel factors in fractions?
3. Can you explain how to factor differences of squares, like \(x^2 - 4\)?
4. What are some restrictions that arise when simplifying rational expressions?
5. How can you determine the domain of rational functions like \( \frac{1}{x + 2} \)?

**Tip:** Always check for restrictions when simplifying, especially when cancelling factors in a fraction.

This solution provides a step-by-step simplification of the rational expression 2x^2 - 4x over 2x^3 - 8x. Learn how to factor polynomials, apply the difference of squares, and cancel common factors. Suitable for students in Grades 8-10.

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