Let's solve the problem step by step. The given expression is:

$\frac{x^2 - 9}{x^2 - 4x - 21} \div \frac{x + 5}{x - 7}$

Step 1: Division of rational expressions

When dividing rational expressions, we multiply by the reciprocal of the second expression. Thus, the expression becomes:

$\frac{x^2 - 9}{x^2 - 4x - 21} \times \frac{x - 7}{x + 5}$

Step 2: Factor each expression

Now, let's factor both the numerator and denominator of each fraction where possible.

1. $x^2 - 9$ is a difference of squares and can be factored as: $x^2 - 9 = (x - 3)(x + 3)$

2. $x^2 - 4x - 21$ factors by finding two numbers that multiply to $-21$ and add to $-4$, which are $-7$ and $3$: $x^2 - 4x - 21 = (x - 7)(x + 3)$

Step 3: Substitute the factored forms into the expression

Now, substitute the factored expressions back into the equation:

$\frac{(x - 3)(x + 3)}{(x - 7)(x + 3)} \times \frac{x - 7}{x + 5}$

Step 4: Cancel out common terms

In the expression, the following terms can be canceled:

• $(x + 3)$ in the numerator and denominator
• $(x - 7)$ in the numerator and denominator

After canceling the common terms, we are left with:

$\frac{x - 3}{x + 5}$

Final Answer:

The simplified expression is:

$\frac{x - 3}{x + 5}$

This is the reduced rational expression.

Would you like more details on any step? Here are some related questions for practice:

1. How do you factor quadratic expressions?
2. What is the difference of squares, and how do you recognize it?
3. How do you divide rational expressions in general?
4. Can you provide another example of simplifying a rational expression?
5. What are the common mistakes when simplifying rational expressions?

Tip: Always look for common factors that can be canceled out when simplifying rational expressions.