1. **Problem:** Evaluate the integral $$\int \left(2x^3 - 3x^2 + 6x - \frac{3}{x^4} + \frac{2}{x}\right) dx$$. 2. **Formula and rules:** The integral of a sum is the sum of the integrals. Use the power rule for integration: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ for $$n \neq -1$$, and $$\int \frac{1}{x} dx = \ln|x| + C$$. 3. **Rewrite terms:** $$\int 2x^3 dx - \int 3x^2 dx + \int 6x dx - \int 3x^{-4} dx + \int 2x^{-1} dx$$ 4. **Integrate each term:** - $$\int 2x^3 dx = 2 \cdot \frac{x^{4}}{4} = \frac{1}{2}x^{4}$$ - $$\int 3x^2 dx = 3 \cdot \frac{x^{3}}{3} = x^{3}$$ - $$\int 6x dx = 6 \cdot \frac{x^{2}}{2} = 3x^{2}$$ - $$\int 3x^{-4} dx = 3 \cdot \frac{x^{-3}}{-3} = -x^{-3}$$ - $$\int 2x^{-1} dx = 2 \ln|x|$$ 5. **Combine results:** $$\frac{1}{2}x^{4} - x^{3} + 3x^{2} - x^{-3} + 2 \ln|x| + C$$ 6. **Final answer:** $$\boxed{\frac{1}{2}x^{4} - x^{3} + 3x^{2} - \frac{1}{x^{3}} + 2 \ln|x| + C}$$