Solve the inequality. (Enter your answer using interval notation.) \[ -x^{3}+6 x^{2}+4 x>24 \]

To solve the inequality \(-x^{3} + 6x^{2} + 4x > 24\), first, we rearrange it to set it to zero: \[ -x^{3} + 6x^{2} + 4x - 24 > 0 \] Next, we can factor or graph the left-hand side to find the roots. To identify the critical points, we find where \(-x^{3} + 6x^{2} + 4x - 24 = 0\). After using numerical methods or graphing/software tools, we find the roots to be approximately \(x \approx -2\), \(x \approx 2\), and \(x \approx 6\). Now, we can test intervals based on these critical points: 1. \( (-\infty, -2) \) 2. \( (-2, 2) \) 3. \( (2, 6) \) 4. \( (6, \infty) \) By choosing test points in each interval, we can determine where the expression is positive: - In \( (-\infty, -2) \), choose \(x = -3\): \[ -(-3)^3 + 6(-3)^2 + 4(-3) - 24 = 27 + 54 - 12 - 24 = 45 > 0 \quad (\text{Positive}) \] - In \( (-2, 2) \), choose \(x = 0\): \[ -(0)^3 + 6(0)^2 + 4(0) - 24 = -24 < 0 \quad (\text{Negative}) \] - In \( (2, 6) \), choose \(x = 4\): \[ -(4)^3 + 6(4)^2 + 4(4) - 24 = -64 + 96 + 16 - 24 = 24 > 0 \quad (\text{Positive}) \] - In \( (6, \infty) \), choose \(x = 7\): \[ -(7)^3 + 6(7)^2 + 4(7) - 24 = -343 + 294 + 28 - 24 = -45 < 0 \quad (\text{Negative}) \] From this testing, we see the expression is positive in the intervals \( (-\infty, -2) \) and \( (2, 6) \). Since we want to exclude where the inequality is equal to zero, our final solution in interval notation is: \[ (-\infty, -2) \cup (2, 6) \]