To simplify the expression

$\frac{x^2 + 6x + 9}{x^2 + 7x + 12},$

we start by factoring both the numerator and the denominator.

Step 1: Factor the numerator

The numerator is $x^2 + 6x + 9$. This is a perfect square trinomial, and it factors as:

$x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2.$

Step 2: Factor the denominator

The denominator is $x^2 + 7x + 12$. To factor this, we look for two numbers that multiply to 12 and add to 7. These numbers are 3 and 4. So, the denominator factors as:

$x^2 + 7x + 12 = (x + 3)(x + 4).$

Step 3: Simplify the expression

Now, substitute the factored forms into the original expression:

$\frac{(x + 3)(x + 3)}{(x + 3)(x + 4)}.$

Cancel the common factor $(x + 3)$ from both the numerator and the denominator:

$\frac{x + 3}{x + 4}.$

Final Answer:

The simplified expression is:

$\frac{x + 3}{x + 4}.$

Would you like further details or have any questions?

Related questions:

1. How can I factor more complex quadratic expressions?
2. What are perfect square trinomials, and how are they identified?
3. Can you explain how to simplify rational expressions with higher degrees?
4. How does factoring help in solving quadratic equations?
5. What happens if $x = -3$ in the simplified expression?

Tip:

Always check if the terms in the numerator and denominator have common factors before simplifying rational expressions.