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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.

Simplify the following expression completely: (x^2 + 3x + 2) / (x^2 + 11x + 18)

Learn how to simplify the rational expression (x^2 + 3x + 2) / (x^2 + 11x + 18) by factoring the numerator and denominator. This problem explores key algebraic concepts including factoring quadratics and simplifying rational expressions. Suitable for students in Grades 8-10.

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