To simplify the expression:

$\frac{x^2 + 11x + 18}{x^2 - 2x - 8}$

Step 1: Factor both the numerator and the denominator.

Factor the numerator $(x^2 + 11x + 18)$: Find two numbers that multiply to 18 and add to 11. These numbers are 9 and 2.

$x^2 + 11x + 18 = (x + 9)(x + 2)$

Factor the denominator $(x^2 - 2x - 8)$: Find two numbers that multiply to -8 and add to -2. These numbers are -4 and 2.

$x^2 - 2x - 8 = (x - 4)(x + 2)$

Step 2: Rewrite the expression with factored forms.

$\frac{(x + 9)(x + 2)}{(x - 4)(x + 2)}$

Step 3: Cancel the common factor $(x + 2)$ from both the numerator and the denominator.

$\frac{x + 9}{x - 4}$

Final Answer:

$\frac{x + 9}{x - 4}$

This is the simplified form of the expression.

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Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you factor quadratic expressions?
2. What happens if there's no common factor to cancel out in a rational expression?
3. Can you explain how the quadratic formula could be used in this case?
4. What restrictions would exist in the domain of the simplified expression?
5. How does this simplification change if a different factorization is required?

Tip: Always check for restrictions on the variable, like $x \neq -2$ and $x \neq 4$ here, since they would make the denominator zero!