The expression you've provided is:

\[
\frac{(p-2)(2p+3)}{2pq+3q}
\]

This is a rational expression, where both the numerator and denominator are polynomials. 

### Simplification Steps:

1. **Factor Common Terms (if possible):**
   - Check if there are any common factors in the numerator and the denominator that can be canceled out. In this case, no common factors are immediately apparent between \((p-2)(2p+3)\) and \(2pq+3q\).

2. **Check for further simplification:**
   - You can factor out \(q\) from the denominator to get:
   \[
   \frac{(p-2)(2p+3)}{q(2p+3)}
   \]
   - Notice now that \((2p+3)\) is a common factor in both the numerator and denominator.

3. **Cancel the Common Factor:**
   - You can cancel the \((2p+3)\) from the numerator and the denominator:
   \[
   \frac{p-2}{q}
   \]

So, the simplified form of the expression is:

\[
\boxed{\frac{p-2}{q}}
\]

Would you like any further details or have any questions?

### Related Questions:
1. How would the expression change if \(p = 1\)?
2. What happens if \(q = 0\) in the expression?
3. How does this expression behave as \(p\) becomes very large?
4. Can the denominator ever be zero? If so, for what values of \(p\) and \(q\)?
5. What if the original expression had a different denominator, say \(2p + 3q\)?

**Tip:** When simplifying expressions, always look for common factors in the numerator and denominator to cancel out, but be careful not to cancel terms that aren't factors.

Learn how to simplify the rational expression (p-2)(2p+3)/(2pq+3q) step-by-step. Understand the process of factoring and canceling common terms. Suitable for high school students learning about rational expressions and factoring.

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