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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.

Each of the regions A, B, and C bounded by the graph of f and the x-axis has area 3. Find the value of ∫_{-4}^{2} [f(x) + 2x + 3] dx.

Solve the definite integral ∫_{-4}^{2} [f(x) + 2x + 3] dx where regions A, B, and C each have an area of 3. Learn how to break down the integral into simpler parts using algebraic properties of integration. Suitable for high school students in Grades 10-12.

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