To solve the given problem, we need to integrate the function $\int (3x^2 + 2x + 3) \, dx$.

Let's perform the integration step-by-step:

1. Integrate $3x^2$: $\int 3x^2 \, dx = \frac{3x^3}{3} = x^3$

2. Integrate $2x$: $\int 2x \, dx = \frac{2x^2}{2} = x^2$

3. Integrate $3$: $\int 3 \, dx = 3x$

Putting it all together, the integral is:

$x^3 + x^2 + 3x + C$

The correct answer is option (d) $x^3 + x^2 + 3x + C$.

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How do you find the indefinite integral of a polynomial function?
2. What are the basic rules of integration?
3. How do constants affect the result of an integration?
4. Can you explain the difference between definite and indefinite integrals?
5. How do you integrate functions that involve products or quotients?

Tip: Remember, when integrating, each term of the function should be integrated separately and constants of integration ($C$) should always be included for indefinite integrals.