1. **State the problem:** We need to find the indefinite integral of the function $12 + 8x - 4x^2$ with respect to $x$. 2. **Recall the integral rules:** - The integral of a constant $a$ is $ax$. - The integral of $x^n$ is $\frac{x^{n+1}}{n+1}$ for $n \neq -1$. 3. **Apply the integral to each term:** $$\int (12 + 8x - 4x^2) \, dx = \int 12 \, dx + \int 8x \, dx - \int 4x^2 \, dx$$ 4. **Integrate each term:** - $\int 12 \, dx = 12x$ - $\int 8x \, dx = 8 \cdot \frac{x^{1+1}}{1+1} = 8 \cdot \frac{x^2}{2} = 4x^2$ - $\int 4x^2 \, dx = 4 \cdot \frac{x^{2+1}}{2+1} = 4 \cdot \frac{x^3}{3} = \frac{4}{3}x^3$ 5. **Combine the results:** $$12x + 4x^2 - \frac{4}{3}x^3 + C$$ 6. **Final answer:** $$\boxed{12x + 4x^2 - \frac{4}{3}x^3 + C}$$