1. The problem is to find the integral $$\int (x-2) \sqrt{2+x} \, dx$$. 2. We use substitution to solve this integral. Let $$u = 2 + x$$, so $$du = dx$$ and $$x = u - 2$$. 3. Substitute into the integral: $$\int (u - 2 - 2) \sqrt{u} \, du = \int (u - 4) u^{\frac{1}{2}} \, du$$. 4. Simplify the integrand: $$\int (u^{\frac{3}{2}} - 4 u^{\frac{1}{2}}) \, du$$. 5. Integrate term by term: $$\int u^{\frac{3}{2}} \, du - 4 \int u^{\frac{1}{2}} \, du = \frac{2}{5} u^{\frac{5}{2}} - 4 \cdot \frac{2}{3} u^{\frac{3}{2}} + C$$. 6. Simplify coefficients: $$\frac{2}{5} u^{\frac{5}{2}} - \frac{8}{3} u^{\frac{3}{2}} + C$$. 7. Substitute back $$u = 2 + x$$: $$\frac{2}{5} (2 + x)^{\frac{5}{2}} - \frac{8}{3} (2 + x)^{\frac{3}{2}} + C$$. Final answer: $$\int (x-2) \sqrt{2+x} \, dx = \frac{2}{5} (2 + x)^{\frac{5}{2}} - \frac{8}{3} (2 + x)^{\frac{3}{2}} + C$$.