1. The problem is to evaluate the indefinite integral $$\int (5x^3 - 2x + 4) \, dx$$. 2. The formula for integrating a power function is $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $n \neq -1$ and $C$ is the constant of integration. 3. Apply the integral to each term separately: $$\int 5x^3 \, dx = 5 \int x^3 \, dx = 5 \cdot \frac{x^{4}}{4} = \frac{5x^4}{4}$$ $$\int (-2x) \, dx = -2 \int x \, dx = -2 \cdot \frac{x^{2}}{2} = -x^2$$ $$\int 4 \, dx = 4x$$ 4. Combine all results and add the constant of integration $C$: $$\int (5x^3 - 2x + 4) \, dx = \frac{5x^4}{4} - x^2 + 4x + C$$ 5. This is the final answer, representing the antiderivative of the given polynomial.