The expression provided in the image is:

$5pq(p^2 - q^2) \div 2p(p + q)$

Let's simplify this step by step.

Step 1: Factorize $p^2 - q^2$

The term $p^2 - q^2$ is a difference of squares, so we can write: $p^2 - q^2 = (p - q)(p + q)$

Substituting this into the expression, we get: $\frac{5pq \cdot (p - q)(p + q)}{2p \cdot (p + q)}$

Step 2: Cancel common terms

The term $p + q$ appears in both the numerator and denominator, so it cancels out: $\frac{5pq \cdot (p - q)}{2p}$

Step 3: Simplify further

The $p$ in the denominator cancels with the $p$ in $5pq$ in the numerator: $\frac{5q \cdot (p - q)}{2}$

Final Simplified Expression:

$\frac{5q(p - q)}{2}$

Would you like a detailed explanation of any step? Let me know!

Here are 5 related questions to deepen your understanding:

1. How do you factorize higher-order polynomials, like $x^3 - y^3$?
2. What are some other cases where terms cancel out during simplification?
3. How does the difference of squares formula apply to other algebraic problems?
4. What is the importance of factoring in simplifying algebraic expressions?
5. How can you check your final simplified answer for correctness?

Tip:

When simplifying fractions, always look for common factors in the numerator and denominator to cancel terms efficiently.