To solve this problem, we are tasked with finding the antiderivative $F(x)$ of the given function $f(x) = 6x + 3$, and we are provided the condition $F(1) = 2$.

Step-by-Step Solution:

1. Find the General Antiderivative: The antiderivative of $f(x) = 6x + 3$ is calculated as: $F(x) = \int (6x + 3) \, dx = 3x^2 + 3x + C,$ where $C$ is the constant of integration.

2. Use the Given Condition $F(1) = 2$: Substitute $x = 1$ and $F(1) = 2$ into the expression for $F(x)$: $F(1) = 3(1)^2 + 3(1) + C = 2.$ Simplify: $3 + 3 + C = 2 \quad \implies \quad 6 + C = 2 \quad \implies \quad C = -4.$

3. Write the Specific Antiderivative: Substitute $C = -4$ back into the general form of $F(x)$: $F(x) = 3x^2 + 3x - 4.$

Final Answer:

The correct option is: $\boxed{F(x) = 3x^2 + 3x - 4.}$

Do you want a more detailed explanation or help with a related problem? Here are some questions that can expand your understanding:

1. What is the general formula for the antiderivative of a polynomial function?
2. How does the constant of integration affect the solution of an indefinite integral?
3. Can you verify this result by differentiating $F(x)$ to get $f(x)$?
4. What is the significance of the initial condition $F(1) = 2$?
5. How does this method apply to other types of functions, such as trigonometric or exponential functions?

Tip: Always verify your solution by differentiating the antiderivative to ensure you retrieve the original function!