Let's first rewrite the given expression in a more standard form:

\[
18xy^3z^2 - 21x^2z^3
\]

### Step-by-Step Solution

1. **Identify the Greatest Common Factor (GCF):**

   - **Coefficients:**  
     The coefficients are 18 and 21.  
     \[
     \text{GCF of } 18 \text{ and } 21 \text{ is } 3.
     \]

   - **Variable \(x\):**  
     The lowest power of \(x\) in both terms is \(x^1\). So, \(x\) is part of the GCF.

   - **Variable \(z\):**  
     The lowest power of \(z\) in both terms is \(z^2\). So, \(z^2\) is part of the GCF.

   - **Variable \(y\):**  
     The variable \(y\) only appears in the first term, so it is not part of the GCF.

   Therefore, the GCF of the expression is:
   \[
   3xz^2
   \]

2. **Factor out the GCF:**

   We can factor out \(3xz^2\) from both terms in the expression:

\[
18xy^3z^2 - 21x^2z^3 = 3xz^2(6y^3 - 7xz)
\]

3. **Verify the Factored Expression:**

   To ensure the factoring is correct, we can distribute \(3xz^2\) back through the parentheses:

   \[
   3xz^2(6y^3) - 3xz^2(7xz) = 18xy^3z^2 - 21x^2z^3
   \]

   This matches the original expression.

### Factored Form:

\[
\boxed{3xz^2(6y^3 - 7xz)}
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you factor an expression with multiple variables?
2. What are the steps to determine the greatest common factor (GCF) of a polynomial?
3. How can you check if your factoring is correct?
4. What strategies can you use when factoring more complex polynomials?
5. What is the significance of factoring in algebraic simplification?

### Tip:

Always check each term carefully for the GCF, including coefficients and each variable, to ensure accurate factoring.

Learn how to factor the expression 18xy^3z^2 - 21x^2z^3 step-by-step. Understand the concept of Greatest Common Factor (GCF) and how to apply it in polynomial expressions. Suitable for high school students learning advanced algebra.

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