The expression you provided is:

\[
\frac{2x^2 - 10x}{6x - 30}
\]

### Step-by-Step Simplification:

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

   - The numerator is \( 2x^2 - 10x \). You can factor out the common factor of \( 2x \):
     \[
     2x^2 - 10x = 2x(x - 5)
     \]

   - The denominator is \( 6x - 30 \). You can factor out the common factor of \( 6 \):
     \[
     6x - 30 = 6(x - 5)
     \]

2. **Rewrite the expression:**

   The expression now becomes:
   \[
   \frac{2x(x - 5)}{6(x - 5)}
   \]

3. **Cancel the common factor** \( (x - 5) \) from the numerator and the denominator (as long as \( x \neq 5 \), to avoid division by zero):
   \[
   \frac{2x(x - 5)}{6(x - 5)} = \frac{2x}{6}
   \]

4. **Simplify the fraction** \( \frac{2x}{6} \) to \( \frac{x}{3} \).

Thus, the simplified form of the expression is:
\[
\frac{x}{3}
\]

### Final Answer:
\[
\frac{2x^2 - 10x}{6x - 30} = \frac{x}{3} \quad \text{(as long as } x \neq 5)
\]

Would you like more details, or do you have any questions?

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Here are 5 related questions to explore further:

1. How do you simplify more complex rational expressions?
2. What happens if \( x = 5 \) in this expression?
3. Can you apply this simplification method to trinomials in both the numerator and denominator?
4. How do you deal with quadratic equations in the denominator?
5. What are the rules for factoring polynomials effectively?

**Tip:** Always look for common factors in both the numerator and denominator to simplify rational expressions.

This page provides a step-by-step solution to simplify the rational expression 2x^2 - 10x / 6x - 30. Learn how to factor polynomials and cancel common factors to reduce the expression. The solution also explains the division by zero rule. Suitable for students in Grades 9-11.

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