The problem in the image you provided appears to involve solving the integral:

$\int \frac{3x^2 + 2x + 1}{x + 2} \, dx$

To solve this, we can start by simplifying the integrand, possibly using polynomial division or substitution methods. Let me walk you through the process:

1. Polynomial Division: First, divide $3x^2 + 2x + 1$ by $x + 2$ to simplify the integrand.

   Dividing $3x^2 + 2x + 1$ by $x + 2$ gives:

$\frac{3x^2 + 2x + 1}{x + 2} = 3x - 4 + \frac{9}{x+2}$

So, the integral becomes:

$\int \left( 3x - 4 + \frac{9}{x+2} \right) \, dx$

2. Integration: Now, we integrate each term separately:

   $\int 3x \, dx = \frac{3x^2}{2}$

   $\int -4 \, dx = -4x$

   $\int \frac{9}{x+2} \, dx = 9 \ln |x+2|$

3. Combine the Results: After integrating each term, we combine them:

$\frac{3x^2}{2} - 4x + 9 \ln |x+2| + C$

Therefore, the answer is:

$\frac{3x^2}{2} - 4x + 9 \ln |x+2| + C$

Since the choices in your image do not exactly match this form, you might need to review the possible numerical constants or format based on specific limits or conditions provided in your problem.

Let me know if you'd like further clarification on any of the steps or if you have additional questions!

Would you like to explore the following?

1. How to handle integration with limits (definite integrals)?
2. A deeper explanation of polynomial division in integration.
3. Solving similar integrals with different polynomial functions.
4. Understanding natural logarithms in integrals.
5. The method of substitution in integration.

Tip: When working with rational functions, always check if polynomial division can simplify the integrand before attempting other methods.