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The expression you have provided is:

\[
\frac{u^2 + 9u + 18}{u + 6}
\]

### To simplify this expression:

1. **Factor the numerator** \( u^2 + 9u + 18 \):
   - We need two numbers that multiply to 18 (the constant term) and add up to 9 (the coefficient of \( u \)).
   - These numbers are 3 and 6 because \( 3 \times 6 = 18 \) and \( 3 + 6 = 9 \).

So, we can rewrite the numerator as:

\[
u^2 + 9u + 18 = (u + 3)(u + 6)
\]

2. **Rewrite the entire expression**:

\[
\frac{(u + 3)(u + 6)}{u + 6}
\]

3. **Cancel out the common factor** \( (u + 6) \) from the numerator and denominator, assuming \( u \neq -6 \) (since division by zero is undefined).

Thus, the simplified expression is:

\[
u + 3
\]

### Final Answer:
The simplified form of the given expression is \( u + 3 \).

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Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:

1. How do you factor a quadratic trinomial in general?
2. What happens if \( u = -6 \) in this problem?
3. How would the problem change if the quadratic had no real factors?
4. Can you explain how common factors in expressions are canceled?
5. How would you simplify a fraction if there were no common factors?

**Tip:** Always check for any restrictions in the variable when simplifying rational expressions to avoid undefined terms.

Learn how to simplify the rational expression (u^2 + 9u + 18) / (u + 6) by factoring the quadratic in the numerator and canceling common factors. This example covers algebraic concepts including factoring quadratics and simplifying rational expressions. Suitable for students in Grades 8-10.

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