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The problem is asking for the most general antiderivative (or indefinite integral) of the given polynomial expression:

\[
\int \left(40x^3 + 12x^2 - 9x + 14 \right) dx
\]

We can solve this by integrating each term separately.

### Step-by-step integration:
1. **Integrating \( 40x^3 \):**
   \[
   \int 40x^3 \, dx = 40 \cdot \frac{x^4}{4} = 10x^4
   \]

2. **Integrating \( 12x^2 \):**
   \[
   \int 12x^2 \, dx = 12 \cdot \frac{x^3}{3} = 4x^3
   \]

3. **Integrating \( -9x \):**
   \[
   \int -9x \, dx = -9 \cdot \frac{x^2}{2} = -\frac{9x^2}{2}
   \]

4. **Integrating \( 14 \):**
   \[
   \int 14 \, dx = 14x
   \]

Now, combining all these results, we get:
\[
10x^4 + 4x^3 - \frac{9x^2}{2} + 14x + C
\]

This matches answer **D** from the options.

### Final Answer: D

Would you like further details on any step? Here are five related questions:

1. How do you handle antiderivatives with negative exponents?
2. What is the difference between definite and indefinite integrals?
3. Why is the constant of integration important?
4. How can you verify the antiderivative by differentiating?
5. What happens when you integrate exponential or logarithmic functions?

**Tip:** Always check the degree of each term during integration—add 1 to the exponent and divide by the new exponent.

Find the most general antiderivative of (40x^3 + 12x^2 - 9x + 14) dx

Learn how to find the most general antiderivative of the polynomial function 40x^3 + 12x^2 - 9x + 14 with a detailed step-by-step solution. This calculus problem covers key integration concepts, including the power rule for integration and the fundamental theorem of calculus. Suitable for students in Grades 11-12.

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