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The given expression to factor completely is:

\[
3n^2 - 15n + 18
\]

### Step 1: Factor out the greatest common factor (GCF).
The GCF of the terms \( 3n^2 \), \( -15n \), and \( 18 \) is 3. So, factor out 3:

\[
3(n^2 - 5n + 6)
\]

### Step 2: Factor the quadratic expression \( n^2 - 5n + 6 \).
Now, we factor the quadratic part \( n^2 - 5n + 6 \). We need to find two numbers that multiply to 6 (the constant term) and add to -5 (the coefficient of \( n \)).

The numbers -2 and -3 work because:

\[
-2 \times -3 = 6 \quad \text{and} \quad -2 + (-3) = -5
\]

So, we can factor \( n^2 - 5n + 6 \) as:

\[
n^2 - 5n + 6 = (n - 2)(n - 3)
\]

### Step 3: Combine everything.
Now substitute this back into the expression:

\[
3(n - 2)(n - 3)
\]

Thus, the completely factored form of the expression is:

\[
\boxed{3(n - 2)(n - 3)}
\]

Would you like further explanation or any details on this process?

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Here are 5 related questions to expand your understanding:
1. How do you factor quadratic expressions where the leading coefficient is not 1?
2. How can you factor trinomials with complex roots?
3. What is the difference between factoring by grouping and factoring by trial and error?
4. How do you recognize when an expression can no longer be factored?
5. What is the role of the discriminant in determining factorability?

**Tip:** When factoring quadratic trinomials, always check if a common factor can be factored out first, as it simplifies the process.

Discover how to factor the quadratic expression 3n^2 - 15n + 18 completely using the greatest common factor (GCF) and the product-sum method. This step-by-step guide is suitable for students in Grades 9-10.

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