The expression you provided is:

\[
6a^2b + 12ab^2 - 18b^3
\]

To factor this expression, we look for the greatest common factor (GCF) of all the terms.

### Step 1: Identify the GCF of the terms.

The coefficients are 6, 12, and 18. The GCF of these numbers is 6.

For the variables, the terms contain both \(a\) and \(b\), but \(a\) only appears in the first two terms. The lowest power of \(a\) is \(a\), and the lowest power of \(b\) is \(b\).

Thus, the GCF of the entire expression is \(6b\).

### Step 2: Factor out the GCF.

Now, divide each term by \(6b\):

\[
6a^2b \div 6b = a^2
\]
\[
12ab^2 \div 6b = 2ab
\]
\[
18b^3 \div 6b = 3b^2
\]

Now, rewrite the expression as:

\[
6b(a^2 + 2ab - 3b^2)
\]

### Step 3: Factor the quadratic expression \(a^2 + 2ab - 3b^2\).

We need to factor \(a^2 + 2ab - 3b^2\). To do this, find two numbers that multiply to give \(-3b^2\) and add to give \(2ab\). The numbers are \(3b\) and \(-b\).

Thus, we factor it as:

\[
(a + 3b)(a - b)
\]

### Final factored form:

\[
6b(a + 3b)(a - b)
\]

Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. How do you find the GCF of more complex expressions?
2. What is the process for factoring trinomials?
3. How would you handle factoring if there were more variables involved?
4. Can this factoring technique be applied to higher-degree polynomials?
5. How would the expression change if it were a difference of squares?

**Tip:** Always look for the greatest common factor first when factoring an expression.

Learn how to factor the expression 6a²b + 12ab² - 18b³ step by step. This solution includes identifying the greatest common factor (GCF) and factoring a quadratic expression, with a detailed explanation suitable for Grades 9-10.

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