To find the antiderivative (also known as the indefinite integral) of the function \( 4x^2 + 2x - 5 \), we integrate each term individually:

\[
\int (4x^2 + 2x - 5) \, dx
\]

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

2. **Integrate \( 2x \)**:
   \[
   \int 2x \, dx = 2 \cdot \frac{x^{2}}{2} = x^2
   \]

3. **Integrate \( -5 \)**:
   \[
   \int -5 \, dx = -5x
   \]

Now, combine the results:

\[
\frac{4x^3}{3} + x^2 - 5x + C
\]

Where \( C \) is the constant of integration.

### Final Answer:
\[
\int (4x^2 + 2x - 5) \, dx = \frac{4x^3}{3} + x^2 - 5x + C
\]

Would you like to see more details or have any other questions?

### Related Questions:
1. What is the antiderivative of \( 4x^3 - x^2 + 6 \)?
2. How do you find the antiderivative of a trigonometric function like \( \sin(x) \)?
3. Can you explain the process of finding a definite integral for a polynomial?
4. What are some common mistakes when calculating antiderivatives?
5. How do you determine the constant of integration \( C \) in specific cases?

### Tip:
When integrating polynomials, remember to increase the exponent by 1 and divide by the new exponent for each term.

Learn how to find the antiderivative of the polynomial function 4x^2 + 2x - 5 step-by-step. This problem involves integration of polynomial functions and understanding the constant of integration. Suitable for students in Grades 10-12.

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