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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