1. **State the problem:** We need to verify and fill in the blanks for the integral $$\int \frac{16x - 24x^2}{x^4} \, dx = -16 x^{-2} + 24 x^{-1} + C$$ 2. **Rewrite the integrand:** Simplify the integrand by dividing each term by $x^4$: $$\frac{16x}{x^4} - \frac{24x^2}{x^4} = 16x^{1-4} - 24x^{2-4} = 16x^{-3} - 24x^{-2}$$ 3. **Set up the integral:** $$\int (16x^{-3} - 24x^{-2}) \, dx$$ 4. **Integrate each term using the power rule:** Recall the power rule for integration: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$ - For $16x^{-3}$: $$16 \int x^{-3} \, dx = 16 \cdot \frac{x^{-3+1}}{-3+1} = 16 \cdot \frac{x^{-2}}{-2} = -8 x^{-2}$$ - For $-24x^{-2}$: $$-24 \int x^{-2} \, dx = -24 \cdot \frac{x^{-2+1}}{-2+1} = -24 \cdot \frac{x^{-1}}{-1} = 24 x^{-1}$$ 5. **Combine the results:** $$-8 x^{-2} + 24 x^{-1} + C$$ 6. **Check the given answer:** The given answer is $$-16 x^{-2} + 24 x^{-1} + C$$ The coefficient of $x^{-2}$ differs. Let's verify the integration step carefully. 7. **Recalculate the first integral term:** $$16 \int x^{-3} \, dx = 16 \cdot \frac{x^{-2}}{-2} = -8 x^{-2}$$ The calculation is correct. So the coefficient should be $-8$, not $-16$. 8. **Conclusion:** The correct integral is $$-8 x^{-2} + 24 x^{-1} + C$$ Therefore, the blanks should be filled as: - First blank: $-8$ - Second blank: $24$ - Third blank (exponent): $-1$ **Note:** The given first blank $-16$ is incorrect based on the integral calculation. --- **Final answer:** $$\int \frac{16x - 24x^2}{x^4} \, dx = -8 x^{-2} + 24 x^{-1} + C$$