1. **State the problem:** We need to evaluate the integral $$\int \sqrt[3]{x} \left(4 - \sqrt{x} + \frac{2}{x \sqrt[3]{x}}\right) \, dx.$$\n\n2. **Rewrite the integrand using exponents:** Recall that $\sqrt[3]{x} = x^{1/3}$ and $\sqrt{x} = x^{1/2}$. Also, $\frac{2}{x \sqrt[3]{x}} = 2 x^{-1} x^{-1/3} = 2 x^{-4/3}$. So the integrand becomes:\n$$x^{1/3} \left(4 - x^{1/2} + 2 x^{-4/3}\right) = 4 x^{1/3} - x^{1/3 + 1/2} + 2 x^{1/3 - 4/3}.$$\n\n3. **Simplify the exponents:**\n- $1/3 + 1/2 = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}$\n- $1/3 - 4/3 = -1$\nSo the integrand is:\n$$4 x^{1/3} - x^{5/6} + 2 x^{-1}.$$\n\n4. **Rewrite the integral:**\n$$\int \left(4 x^{1/3} - x^{5/6} + 2 x^{-1}\right) dx = \int 4 x^{1/3} dx - \int x^{5/6} dx + \int 2 x^{-1} dx.$$\n\n5. **Integrate each term separately:**\n- For $\int x^{n} dx = \frac{x^{n+1}}{n+1} + C$ if $n \neq -1$.\n- For $\int x^{-1} dx = \ln|x| + C$.\n\nCalculate each integral:\n- $\int 4 x^{1/3} dx = 4 \cdot \frac{x^{1/3 + 1}}{1/3 + 1} = 4 \cdot \frac{x^{4/3}}{4/3} = 4 \cdot \frac{3}{4} x^{4/3} = 3 x^{4/3}$.\n- $\int x^{5/6} dx = \frac{x^{5/6 + 1}}{5/6 + 1} = \frac{x^{11/6}}{11/6} = \frac{6}{11} x^{11/6}$.\n- $\int 2 x^{-1} dx = 2 \ln|x|$.\n\n6. **Combine the results:**\n$$3 x^{4/3} - \frac{6}{11} x^{11/6} + 2 \ln|x| + C.$$\n\n**Final answer:**\n$$\boxed{3 x^{4/3} - \frac{6}{11} x^{11/6} + 2 \ln|x| + C}.$$