To simplify the expression $\frac{x^2 - 9}{x^2 - 7x + 12}$, we can begin by factoring both the numerator and the denominator.

Step 1: Factor the numerator

The numerator is $x^2 - 9$. This is a difference of squares, so it factors as: $x^2 - 9 = (x - 3)(x + 3)$

Step 2: Factor the denominator

The denominator is $x^2 - 7x + 12$. To factor this quadratic expression, we need two numbers that multiply to give 12 and add to give -7. These numbers are -3 and -4, so we can factor the denominator as: $x^2 - 7x + 12 = (x - 3)(x - 4)$

Step 3: Simplify the expression

Now, substitute the factored forms of the numerator and denominator into the original expression: $\frac{(x - 3)(x + 3)}{(x - 3)(x - 4)}$ Since $(x - 3)$ appears in both the numerator and denominator, we can cancel it out (as long as $x \neq 3$, to avoid division by zero). This gives us the simplified expression: $\frac{x + 3}{x - 4}$

Final Answer:

$\frac{x + 3}{x - 4}, \quad \text{where} \quad x \neq 3 \quad \text{and} \quad x \neq 4 \, (\text{to avoid division by zero}).$

Would you like more details on any step?

Here are 5 related questions:

1. How do you factor a quadratic expression like $x^2 - 7x + 12$?
2. What is the difference of squares and how do you recognize it?
3. Why do we have to exclude $x = 3$ and $x = 4$ from the solution?
4. How do you solve rational expressions in general?
5. What happens when you cannot cancel any factors in a rational expression?

Tip: Always check for excluded values after canceling terms in rational expressions to avoid dividing by zero!