Question: User: \int -21(-x^2 - x + 2) \, dx
1. **State the problem:** We need to evaluate the integral $$\int -21(-x^2 - x + 2) \, dx$$.
2. **Rewrite the integral:** Distribute the constant $$-21$$ inside the integral:
$$\int -21(-x^2 - x + 2) \, dx = \int (21x^2 + 21x - 42) \, dx$$
3. **Use the power rule for integration:** Recall that $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ for any real number $$n \neq -1$$.
4. **Integrate each term separately:**
$$\int 21x^2 \, dx = 21 \int x^2 \, dx = 21 \cdot \frac{x^{3}}{3} = 7x^{3}$$
$$\int 21x \, dx = 21 \int x \, dx = 21 \cdot \frac{x^{2}}{2} = \frac{21}{2}x^{2}$$
$$\int -42 \, dx = -42x$$
5. **Combine the results:**
$$7x^{3} + \frac{21}{2}x^{2} - 42x + C$$
6. **Final answer:**
$$\int -21(-x^2 - x + 2) \, dx = 7x^{3} + \frac{21}{2}x^{2} - 42x + C$$