1. **Problem statement:** Calculate the integral $$\int \sqrt{x} (x + 2) \, dx$$. 2. **Rewrite the integrand:** Recall that $$\sqrt{x} = x^{\frac{1}{2}}$$, so the integrand becomes $$x^{\frac{1}{2}} (x + 2) = x^{\frac{1}{2}} \cdot x + x^{\frac{1}{2}} \cdot 2 = x^{\frac{3}{2}} + 2x^{\frac{1}{2}}$$. 3. **Integral formula:** Use the power rule for integration: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$, valid for $$n \neq -1$$. 4. **Integrate each term separately:** $$\int x^{\frac{3}{2}} \, dx = \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = \frac{2}{5} x^{\frac{5}{2}}$$ $$\int 2x^{\frac{1}{2}} \, dx = 2 \int x^{\frac{1}{2}} \, dx = 2 \cdot \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} = 2 \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = 2 \cdot \frac{2}{3} x^{\frac{3}{2}} = \frac{4}{3} x^{\frac{3}{2}}$$ 5. **Combine results:** $$\int \sqrt{x} (x + 2) \, dx = \frac{2}{5} x^{\frac{5}{2}} + \frac{4}{3} x^{\frac{3}{2}} + C$$ 6. **Final answer:** $$\boxed{\frac{2}{5} x^{\frac{5}{2}} + \frac{4}{3} x^{\frac{3}{2}} + C}$$