The problem provides a graph of a function $f(x)$ with three shaded regions $A$, $B$, and $C$, all bounded by the graph of $f$ and the x-axis. Each region has an area of 3. The task is to evaluate the integral:

$\int_{-4}^{2} \left( f(x) + 2x + 3 \right) \, dx$

Step-by-Step Solution:

1. Separate the Integral: We can split the given integral into two parts:

   $\int_{-4}^{2} \left( f(x) + 2x + 3 \right) \, dx = \int_{-4}^{2} f(x) \, dx + \int_{-4}^{2} (2x + 3) \, dx$

2. Evaluate the First Integral:
   We are given the areas of the regions bounded by the function $f(x)$:

   • The area of region $A$ is 3.
   • The area of region $B$ is 3.
   • The area of region $C$ is 3.

   Since region $A$ is below the x-axis and regions $B$ and $C$ are above the x-axis, the integral of $f(x)$ from $-4$ to 2 can be written as:

   $\int_{-4}^{2} f(x) \, dx = -3 + 3 = 0$

3. Evaluate the Second Integral: Now, compute the integral of $2x + 3$ over the interval $[-4, 2]$:

   $\int_{-4}^{2} (2x + 3) \, dx = \int_{-4}^{2} 2x \, dx + \int_{-4}^{2} 3 \, dx$

   The first part is:

   $\int_{-4}^{2} 2x \, dx = \left[ x^2 \right]_{-4}^{2} = (2^2) - (-4)^2 = 4 - 16 = -12$

   The second part is:

   $\int_{-4}^{2} 3 \, dx = 3 \times \left( 2 - (-4) \right) = 3 \times 6 = 18$

   So, the total for this integral is:

   $-12 + 18 = 6$

4. Final Answer: Adding the results from both integrals, we get:

   $0 + 6 = 6$

Thus, the value of the integral is $\boxed{6}$.

Would you like any further details or explanations? Here are some related questions you might find useful:

1. What if the function $f(x)$ had different areas for the regions A, B, and C?
2. How do we compute the integral of a function that crosses the x-axis at multiple points?
3. Why do we subtract the area for region A but add it for regions B and C?
4. Can we directly integrate functions involving constants like $2x + 3$?
5. How would this problem change if the limits of integration were different?

Tip: When dealing with areas under a curve, remember that regions below the x-axis contribute negative values to the integral, while regions above contribute positive values.