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The given expression is:

\[
\frac{4a \times 3a^3}{6a^2}
\]

### Step-by-step simplification:

1. Multiply the terms in the numerator:
   \[
   4a \times 3a^3 = 12a^4
   \]
   So the expression becomes:
   \[
   \frac{12a^4}{6a^2}
   \]

2. Simplify the coefficients:
   \[
   \frac{12}{6} = 2
   \]
   So now it becomes:
   \[
   \frac{12a^4}{6a^2} = 2a^{4-2} = 2a^2
   \]

The fully simplified expression is \( 2a^2 \).

Thus, the correct answer is **\( 2a^2 \)**, which matches the third option.

Would you like further clarification or additional details?

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Here are 5 related questions for practice:
1. How do you simplify \(\frac{8a^5 \times 2a^2}{4a^3}\)?
2. What is the rule for dividing terms with the same base and different exponents?
3. How would you simplify \(\frac{15b^6 \times 5b^2}{25b^3}\)?
4. If you multiply \(4x \times 2x^3\), what do you get?
5. How do you simplify expressions with negative exponents, like \(\frac{4a^{-2}}{a^3}\)?

**Tip:** Always handle coefficients and variables separately when simplifying expressions involving exponents.

Learn how to simplify the algebraic expression (4a × 3a^3) / 6a^2 with step-by-step guidance. This example covers key concepts in algebra and exponent rules, suitable for students in Grades 7-9.

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