1. **State the problem:** We need to find the indefinite integral $$\int (1 + x)(1 + 2x) \, dx$$. 2. **Expand the integrand:** Use the distributive property to multiply the two binomials: $$ (1 + x)(1 + 2x) = 1 \cdot 1 + 1 \cdot 2x + x \cdot 1 + x \cdot 2x = 1 + 2x + x + 2x^2 = 1 + 3x + 2x^2 $$ 3. **Rewrite the integral:** $$ \int (1 + 3x + 2x^2) \, dx $$ 4. **Integrate term-by-term:** Recall the power rule for integration: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ for any real number $n \neq -1$. Apply this to each term: - $$\int 1 \, dx = x$$ - $$\int 3x \, dx = 3 \cdot \frac{x^2}{2} = \frac{3x^2}{2}$$ - $$\int 2x^2 \, dx = 2 \cdot \frac{x^3}{3} = \frac{2x^3}{3}$$ 5. **Combine the results:** $$ x + \frac{3x^2}{2} + \frac{2x^3}{3} + C $$ 6. **Final answer:** $$ \boxed{x + \frac{3x^2}{2} + \frac{2x^3}{3} + C} $$ This is the antiderivative of the given function.