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The problem presented asks to divide the following expression:

\[
\frac{12x^3y^2 - 6x^2y + 4xy}{2xy}
\]

### Step-by-Step Solution:

1. **Separate the terms**:
   Each term in the numerator is divided by the denominator \(2xy\).

\[
\frac{12x^3y^2}{2xy} - \frac{6x^2y}{2xy} + \frac{4xy}{2xy}
\]

2. **Simplify each term**:
   - For the first term:
   \[
   \frac{12x^3y^2}{2xy} = \frac{12}{2} \cdot \frac{x^3}{x} \cdot \frac{y^2}{y} = 6x^2y
   \]
   - For the second term:
   \[
   \frac{6x^2y}{2xy} = \frac{6}{2} \cdot \frac{x^2}{x} \cdot \frac{y}{y} = 3x
   \]
   - For the third term:
   \[
   \frac{4xy}{2xy} = \frac{4}{2} \cdot \frac{xy}{xy} = 2
   \]

3. **Final Expression**:
   After simplifying each term, the result is:
   \[
   6x^2y - 3x + 2
   \]

### Answer:
The correct option is:
\[
\boxed{6x^2y - 3x + 2}
\]

Would you like further details or explanations? Here are 5 related questions for practice:

1. How would the answer change if the denominator was \(3xy\) instead of \(2xy\)?
2. Can you express each step algebraically and describe why the powers of \(x\) and \(y\) reduce as they do?
3. What happens if one of the terms in the numerator is missing, e.g., \(12x^3y^2 - 4xy\)?
4. How would you handle this division if the variables were different, such as \(a^3b^2 - ab\)?
5. How would you solve this if you had to factor the numerator first?

**Tip**: Always check if common factors in the numerator and denominator can be canceled before expanding terms!

Learn how to divide the polynomial (12x^3y^2 - 6x^2y + 4xy) by 2xy with a detailed step-by-step solution. This problem involves polynomial division, laws of exponents, and simplification. Suitable for students in Grades 9-12.

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