1. **State the problem:** We need to find the integral $$\int (x+2)(5x-2)^3 \, dx$$. 2. **Use substitution to simplify:** Let $$u = 5x - 2$$. Then, $$\frac{du}{dx} = 5$$ or $$dx = \frac{du}{5}$$. 3. **Express $$x+2$$ in terms of $$u$$:** From $$u = 5x - 2$$, we get $$x = \frac{u + 2}{5}$$. So, $$x + 2 = \frac{u + 2}{5} + 2 = \frac{u + 2}{5} + \frac{10}{5} = \frac{u + 12}{5}$$. 4. **Rewrite the integral in terms of $$u$$:** $$\int (x+2)(5x-2)^3 \, dx = \int \frac{u + 12}{5} u^3 \frac{du}{5} = \int \frac{u + 12}{5} u^3 \cdot \frac{1}{5} du = \int \frac{u^3 (u + 12)}{25} du$$. 5. **Simplify the integrand:** $$\frac{u^3 (u + 12)}{25} = \frac{u^4 + 12u^3}{25}$$. 6. **Integrate term-by-term:** $$\int \frac{u^4 + 12u^3}{25} du = \frac{1}{25} \int (u^4 + 12u^3) du = \frac{1}{25} \left( \frac{u^5}{5} + 12 \cdot \frac{u^4}{4} \right) + C$$. 7. **Simplify the constants:** $$= \frac{1}{25} \left( \frac{u^5}{5} + 3u^4 \right) + C = \frac{u^5}{125} + \frac{3u^4}{25} + C$$. 8. **Substitute back $$u = 5x - 2$$:** $$\int (x+2)(5x-2)^3 \, dx = \frac{(5x - 2)^5}{125} + \frac{3(5x - 2)^4}{25} + C$$. **Final answer:** $$\boxed{\frac{(5x - 2)^5}{125} + \frac{3(5x - 2)^4}{25} + C}$$