1. **State the problem:** Evaluate the integral $$\int (x+1)(3x^2+2) \, dx$$. 2. **Expand the integrand:** Use distributive property: $$ (x+1)(3x^2+2) = x \cdot 3x^2 + x \cdot 2 + 1 \cdot 3x^2 + 1 \cdot 2 = 3x^3 + 2x + 3x^2 + 2 $$ 3. **Rewrite the integral:** $$ \int (3x^3 + 3x^2 + 2x + 2) \, dx $$ 4. **Integrate term-by-term:** - Integral of $$3x^3$$ is $$3 \cdot \frac{x^{4}}{4} = \frac{3}{4}x^{4}$$ - Integral of $$3x^2$$ is $$3 \cdot \frac{x^{3}}{3} = x^{3}$$ - Integral of $$2x$$ is $$2 \cdot \frac{x^{2}}{2} = x^{2}$$ - Integral of $$2$$ is $$2x$$ 5. **Combine results:** $$ \int (x+1)(3x^2+2) \, dx = \frac{3}{4}x^{4} + x^{3} + x^{2} + 2x + C $$ 6. **Explain:** We expanded the product to simplify integration, then applied the power rule for each term, adding the constant of integration $C$ because it is an indefinite integral. **Final answer:** $$ \boxed{\frac{3}{4}x^{4} + x^{3} + x^{2} + 2x + C} $$