Let's factor the expression $12x^3y^2 - 8x^2y^2 + 4xy$.

Step 1: Identify the Greatest Common Factor (GCF)

• Coefficients: The GCF of 12, 8, and 4 is 4.
• Variables:
  • For $x^3$, $x^2$, and $x$, the GCF is $x$.
  • For $y^2$, $y^2$, and $y$, the GCF is $y$.

So, the GCF of the entire expression is $4xy$.

Step 2: Factor out the GCF

Now, factor $4xy$ from each term:

$12x^3y^2 - 8x^2y^2 + 4xy = 4xy(3x^2y - 2xy + 1)$

Final Answer:

The factored form of the expression is:

$4xy(3x^2y - 2xy + 1)$

Would you like further explanation or have any questions?

Here are five related questions to deepen your understanding:

1. How do you factor out the GCF when the expression contains multiple variables?
2. What happens if you overlook the GCF in the first step of factoring?
3. How would the expression change if there were no common factor?
4. Can you apply a different factoring technique if the GCF is 1?
5. What are the next steps if the expression inside the parentheses can be factored further?

Tip: Always double-check your factoring by distributing the GCF back through the expression to ensure it matches the original.