The given integral is:

$\int (3x^2 + 8x - 2) \, dx$

To solve this, we apply the power rule of integration, which states that:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

Now, integrate each term separately:

1. $\int 3x^2 \, dx = 3 \cdot \frac{x^3}{3} = x^3$
2. $\int 8x \, dx = 8 \cdot \frac{x^2}{2} = 4x^2$
3. $\int (-2) \, dx = -2x$

Now, putting it all together:

$x^3 + 4x^2 - 2x + C$

where $C$ is the constant of integration.

Thus, the solution is:

$x^3 + 4x^2 - 2x + C$

Would you like more details or have any further questions?

Here are some related questions:

1. What is the process for finding the integral of a polynomial function?
2. How do we apply the power rule of integration?
3. Can you explain the significance of the constant of integration?
4. What are the common integration rules for different types of functions?
5. How would you handle definite integrals for similar polynomial functions?

Tip: Always remember to include the constant $C$ when solving indefinite integrals.