To solve the problem, we need to perform and simplify the given expression:

$\frac{x^2 + 3x - 4}{x^2 - 9} \div \frac{x + 4}{x - 3}$

Step-by-Step Solution

1. Rewrite the Division as Multiplication: Dividing by a fraction is the same as multiplying by its reciprocal. Thus, we have: $\frac{x^2 + 3x - 4}{x^2 - 9} \times \frac{x - 3}{x + 4}$

2. Factor the Expressions:

   • For the numerator $x^2 + 3x - 4$: $x^2 + 3x - 4 = (x + 4)(x - 1)$
   • For the denominator $x^2 - 9$: $x^2 - 9 = (x + 3)(x - 3)$

   Substituting these factors back into the expression, we get: $\frac{(x + 4)(x - 1)}{(x + 3)(x - 3)} \times \frac{x - 3}{x + 4}$

3. Cancel Common Terms:

   • The $x + 4$ terms in the numerator and denominator cancel out.
   • The $x - 3$ terms in the numerator and denominator also cancel out.

   This simplifies to: $\frac{x - 1}{x + 3}$

Final Answer

The simplified expression is:

$\frac{x - 1}{x + 3}$

Correct Answer Choice

The correct answer is: A. $\frac{x - 1}{x + 3}$

Would you like additional details or have any questions?

Here are some related questions for practice:

1. How would you simplify $\frac{x^2 - 5x + 6}{x^2 - 9} \div \frac{x - 2}{x + 3}$?
2. What is the result of $\frac{2x^2 + 7x + 3}{x^2 - 16} \times \frac{x - 4}{x + 3}$?
3. How do you simplify $\frac{x^2 + 2x - 8}{x^2 - 4} \div \frac{x + 2}{x - 2}$?
4. Can you solve $\frac{x^2 + 6x + 9}{x^2 - 9} \times \frac{x - 3}{x + 3}$?
5. What steps are required to simplify $\frac{x^2 - x - 6}{x^2 - 4x + 3} \div \frac{x - 2}{x - 1}$?

Tip: When simplifying expressions with division, always rewrite the division as multiplication by the reciprocal of the second fraction and look for common factors to cancel out.