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?

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.