1. **State the problem:** Evaluate the integral $$\int (4x^2 - 8x + 1) \, dx$$. 2. **Recall the integral rules:** - The integral of $$x^n$$ is $$\frac{x^{n+1}}{n+1} + C$$ for $$n \neq -1$$. - The integral of a sum is the sum of the integrals. 3. **Apply the integral to each term:** $$\int 4x^2 \, dx = 4 \int x^2 \, dx = 4 \cdot \frac{x^{3}}{3} = \frac{4}{3}x^{3}$$ $$\int (-8x) \, dx = -8 \int x \, dx = -8 \cdot \frac{x^{2}}{2} = -4x^{2}$$ $$\int 1 \, dx = x$$ 4. **Combine all parts:** $$\int (4x^2 - 8x + 1) \, dx = \frac{4}{3}x^{3} - 4x^{2} + x + C$$ 5. **Final answer:** $$\boxed{\frac{4}{3}x^{3} - 4x^{2} + x + C}$$