The problem asks us to find the product or quotient of two rational expressions:

$\frac{4p - 4}{p} \div \frac{5p - 5}{2p^2}.$

Step-by-step Solution:

1. Rewrite the Division as Multiplication: Division of fractions is equivalent to multiplying the first fraction by the reciprocal of the second fraction: $\frac{4p - 4}{p} \div \frac{5p - 5}{2p^2} = \frac{4p - 4}{p} \cdot \frac{2p^2}{5p - 5}.$

2. Factor the Numerators and Denominators: Factor out common terms where possible:

   • In $4p - 4$, factor out 4: $4p - 4 = 4(p - 1)$.
   • In $5p - 5$, factor out 5: $5p - 5 = 5(p - 1)$.

   The expression now becomes: $\frac{4(p - 1)}{p} \cdot \frac{2p^2}{5(p - 1)}.$

3. Cancel Common Factors:

   • $p - 1$ appears in both the numerator and denominator, so cancel $p - 1$.
   • Simplify $p$ in the denominator of the first fraction with $p^2$ in the numerator of the second fraction.

   After canceling, we get: $\frac{4}{1} \cdot \frac{2p}{5}.$

4. Multiply the Remaining Terms: Multiply the numerators and denominators: $\frac{4 \cdot 2p}{1 \cdot 5} = \frac{8p}{5}.$

Final Answer:

$\frac{8p}{5}.$

Would you like me to clarify any steps or go into more detail?

Related Questions:

1. What is the difference between dividing and multiplying rational expressions?
2. How do you factor polynomials with multiple variables?
3. What happens if one of the terms cannot be factored?
4. How can you check your work when simplifying rational expressions?
5. How would the process change if an additional term was added to one of the numerators?

Tip:

Always factor completely before canceling terms to avoid mistakes in simplification.