Let's simplify the given expression step by step:

The expression is:

$\frac{2x^2 - 10x}{x+2} \cdot \frac{6}{2x^2 - 10x}$

Step 1: Factor common terms

Factor the numerator $2x^2 - 10x$ in both fractions:

$2x^2 - 10x = 2x(x - 5)$

So, the expression becomes:

$\frac{2x(x - 5)}{x + 2} \cdot \frac{6}{2x(x - 5)}$

Step 2: Simplify the common factors

Notice that $2x(x - 5)$ appears in both the numerator and denominator, so they cancel out:

$\frac{1}{x + 2} \cdot 6$

Step 3: Multiply what's left

Now multiply the fractions:

$\frac{6}{x + 2}$

Final Answer:

The simplified expression is:

$\frac{6}{x + 2}$

Would you like any further details or explanations?

Here are 5 follow-up questions to deepen understanding:

1. What happens if $x = -2$ in the simplified expression?
2. How would you simplify if the original expression had a different denominator?
3. Can you generalize the process of canceling common factors in other rational expressions?
4. How do you handle complex fractions with polynomials in both numerator and denominator?
5. What is the importance of factoring when simplifying algebraic expressions?

Tip: Always check for common factors before multiplying or dividing expressions—it simplifies the work and reduces errors.