1. Problem: Integrate the polynomial function $$4x^3 + 6x^2 + 2x + 1$$ with respect to $$x$$. 2. Recall the power rule for integration: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$, where $$C$$ is the constant of integration. 3. Apply integration term-by-term: - $$\int 4x^3 dx = 4 \times \frac{x^{4}}{4} = x^4$$ - $$\int 6x^2 dx = 6 \times \frac{x^{3}}{3} = 2x^3$$ - $$\int 2x dx = 2 \times \frac{x^{2}}{2} = x^2$$ - $$\int 1 dx = x$$ 4. Combine all the results and add the constant of integration $$C$$: $$x^4 + 2x^3 + x^2 + x + C$$ 5. Compare with the options given; option a matches the integral. Final answer: $$x^4 + 2x^3 + x^2 + x + C$$