5) Solve the following inequality. State your solution using interval notation. \[ \frac{x^{3}+3 x^{2}-4 x-12}{x+3}>x-2 \]

Step 1 ：First, we bring all terms to one side of the inequality to get \(-x + 2 + \frac{x^{3} + 3x^{2} - 4x - 12}{x + 3} > 0\).

Step 2 ：Next, we factor the polynomial to get \(x^{2} - x - 2 > 0\).

Step 3 ：Then, we find the critical points by setting the polynomial equal to zero. The critical points are -1 and 2.

Step 4 ：These critical points divide the number line into three intervals: \((-\infty, -1)\), \((-1, 2)\), and \((2, \infty)\).

Step 5 ：We test a number from each interval in the original inequality. The test points are -2, 0, and 3.

Step 6 ：The test points -2 and 3 satisfy the inequality, which means the intervals \((-\infty, -1)\) and \((2, \infty)\) are part of the solution. The interval \((-1, 2)\) is not part of the solution because the test point 0 does not satisfy the inequality.

Step 7 ：Finally, the solution to the inequality is \(\boxed{(-\infty, -1) \cup (2, \infty)}\).