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This expression is a complex rational function. Let’s break it down step by step.

The given expression is:

\[
\frac{\frac{4x^2 - 9}{2x + 5}}{\frac{(2x - 3)^2}{4x^2 - 25}}
\]

We can simplify this by following these steps:

1. **Factor any quadratic terms**:
   - \( 4x^2 - 9 \) is a difference of squares, so it can be factored as \( (2x - 3)(2x + 3) \).
   - \( 4x^2 - 25 \) is also a difference of squares, and it factors as \( (2x - 5)(2x + 5) \).

2. **Rewrite the expression** using these factored terms:

\[
\frac{\frac{(2x - 3)(2x + 3)}{2x + 5}}{\frac{(2x - 3)^2}{(2x - 5)(2x + 5)}}
\]

3. **Simplify the complex fraction**:
   - To simplify a complex fraction, multiply by the reciprocal of the denominator of the outer fraction:

\[
\frac{(2x - 3)(2x + 3)}{2x + 5} \times \frac{(2x - 5)(2x + 5)}{(2x - 3)^2}
\]

4. **Cancel common terms**:
   - The term \( (2x + 5) \) cancels out from both the numerator and denominator.
   - One factor of \( (2x - 3) \) cancels out.

This gives us:

\[
\frac{(2x + 3)(2x - 5)}{(2x - 3)}
\]

5. **Final simplified expression**:

\[
\frac{(2x + 3)(2x - 5)}{2x - 3}
\]

This is the simplified form of the given complex rational expression.

Would you like further clarification on any step or help with a different concept? Here are some related questions you might find helpful:

1. How do you factor quadratic expressions like \( 4x^2 - 9 \) and \( 4x^2 - 25 \)?
2. What is the general process for simplifying complex rational expressions?
3. Can you give more examples of simplifying complex fractions?
4. How do you handle higher powers in rational expressions?
5. What happens when a term cancels out in a rational expression?

**Tip**: Always look for common factoring patterns, like the difference of squares, to simplify expressions quickly.

Simplify the complex rational expression \[ \frac{\frac{4x^2 - 9}{2x + 5}}{\frac{(2x - 3)^2}{4x^2 - 25}} \]

Learn how to simplify a complex rational expression involving factoring quadratic expressions and handling nested fractions. This step-by-step guide covers the difference of squares and rules for simplifying complex fractions. Suitable for students in Grades 9-12.

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