The given expression is:

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

To simplify this, we first need to factor the quadratic expression in the numerator $x^2 + 11x + 18$.

Step 1: Factor the numerator

We are looking for two numbers that multiply to give $18$ and add up to $11$. These numbers are $9$ and $2$, so we can factor the quadratic as follows:

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

Step 2: Simplify the expression

Now that the numerator is factored, we rewrite the expression:

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

We can cancel out the common factor $(x + 2)$ from both the numerator and the denominator (as long as $x \neq -2$ to avoid division by zero). This leaves us with:

$x + 9$

Final Answer:

The simplified expression is:

$x + 9$

Let me know if you'd like further clarification or additional details!

Here are some related questions to expand on this concept:

1. What happens if $x = -2$ in this expression?
2. How do you factor a quadratic equation when the leading coefficient is not 1?
3. Can all quadratic expressions be factored? If not, why?
4. How does this simplification process relate to the concept of domain restrictions?
5. What are some real-world applications of simplifying rational expressions?

Tip: Always check for factors that can be canceled in rational expressions, but remember to account for any restrictions on the variable to avoid division by zero.