The expression given is:

$\frac{x^2 + 3x + 2}{x^2 + 11x + 18}$

To simplify this, we will first factor both the numerator and the denominator.

Step 1: Factor the numerator

The numerator is $x^2 + 3x + 2$. We need to find two numbers that multiply to $2$ (the constant term) and add to $3$ (the coefficient of $x$).

Those two numbers are $1$ and $2$, so we can factor the numerator as: $x^2 + 3x + 2 = (x + 1)(x + 2)$

Step 2: Factor the denominator

The denominator is $x^2 + 11x + 18$. We need to find two numbers that multiply to $18$ and add to $11$.

Those two numbers are $2$ and $9$, so we can factor the denominator as: $x^2 + 11x + 18 = (x + 2)(x + 9)$

Step 3: Simplify the expression

Now that both the numerator and denominator are factored, we have:

$\frac{(x + 1)(x + 2)}{(x + 2)(x + 9)}$

We can cancel out the common factor $(x + 2)$ in both the numerator and denominator (as long as $x \neq -2$):

$\frac{x + 1}{x + 9}$

So, the simplified expression is:

$\frac{x + 1}{x + 9}, \quad \text{for} \, x \neq -2$

Would you like further details on any step? Here are 5 follow-up questions you might find useful:

1. How do you factor quadratic expressions systematically?
2. What does it mean to simplify a rational expression?
3. Why is it important to note the restriction $x \neq -2$ in the simplification?
4. Can this method be applied to higher-degree polynomials?
5. How would the graph of this simplified function compare to the original?

Tip: Always check for restrictions on $x$ when simplifying rational expressions, as factors canceled out can indicate values that make the original expression undefined.