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**Problem Statement:**

"Which of the following represents a correct simplification of the given expression?"

Given expression:
\[ \frac{(x^2 + 7x + 10) \div (x^2 - 4)}{(x^2 - x - 20) \div (x^2 - 9)} \]

**Options:**
- a) \( \frac{x+2}{x-3} \)
- b) \( \frac{x+5}{x-3} \)
- c) \( \frac{x-3}{x+2} \)
- d) \( \frac{x-5}{x+3} \)

**Solution:**

1. **Factorize each quadratic expression:**

   - \( x^2 + 7x + 10 \) factors as \( (x+5)(x+2) \).
   - \( x^2 - 4 \) factors as \( (x+2)(x-2) \).
   - \( x^2 - x - 20 \) factors as \( (x-5)(x+4) \).
   - \( x^2 - 9 \) factors as \( (x+3)(x-3) \).

2. **Rewrite the expression:**

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

3. **Simplify the fractions:**

   - Simplify the first fraction: \( \frac{(x+5)(x+2)}{(x+2)(x-2

Which of the following represents a correct simplification of the given expression? Given expression: [(x^2 + 7x + 10) ÷ (x^2 - 4)] ÷ [(x^2 - x - 20) ÷ (x^2 - 9)]

This problem involves simplifying rational expressions with quadratic equations. Learn how to factorize and simplify the given algebraic expression step by step. Suitable for students in Grades 8-10.

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