Let's simplify the expression:

$\frac{x^2 + 14x + 49}{x^2 + 16x + 63}$

Step 1: Factor the numerator and the denominator.

• Numerator: $x^2 + 14x + 49$

This is a perfect square trinomial, and it can be factored as:

$x^2 + 14x + 49 = (x + 7)^2$

• Denominator: $x^2 + 16x + 63$

We need to find two numbers that multiply to 63 and add to 16. These numbers are 9 and 7, so the denominator factors as:

$x^2 + 16x + 63 = (x + 7)(x + 9)$

Step 2: Simplify the expression.

Now substitute the factored forms back into the original expression:

$\frac{(x + 7)^2}{(x + 7)(x + 9)}$

Cancel out one $(x + 7)$ from the numerator and the denominator:

$\frac{x + 7}{x + 9}$

Final Answer:

The simplified expression is:

$\frac{x + 7}{x + 9}$

Would you like more details or have any questions about this solution? Here are five related questions you might consider:

1. How do you factor quadratic expressions in general?
2. Can all quadratic expressions be factored easily?
3. What happens if $x = -9$ in this expression?
4. How do perfect square trinomials differ from other quadratics?
5. How would you graph the simplified expression?

Tip: When simplifying rational expressions, always check for common factors between the numerator and denominator to cancel them out!