To solve the polynomial inequality $-(x-9)(x+1)(x+3) > 0$, let's proceed step by step:

Step 1: Simplify the inequality

We can first distribute the negative sign to simplify the inequality: $-(x-9)(x+1)(x+3) > 0 \quad \text{is the same as} \quad (x-9)(x+1)(x+3) < 0$ Now, we need to solve: $(x-9)(x+1)(x+3) < 0$

Step 2: Identify the critical points

The critical points are found by setting each factor equal to zero: $x - 9 = 0 \quad \Rightarrow \quad x = 9$ $x + 1 = 0 \quad \Rightarrow \quad x = -1$ $x + 3 = 0 \quad \Rightarrow \quad x = -3$ So, the critical points are $x = -3, -1, 9$.

Step 3: Analyze the sign of the expression

We will analyze the sign of the expression in each interval formed by the critical points. These intervals are:

• $(-\infty, -3)$
• $(-3, -1)$
• $(-1, 9)$
• $(9, \infty)$

Pick test points from each interval to determine the sign of the expression $(x-9)(x+1)(x+3)$ in that interval:

1. For $x \in (-\infty, -3)$, choose $x = -4$: $(-4 - 9)(-4 + 1)(-4 + 3) = (-13)(-3)(-1) = -39 \quad \Rightarrow \quad \text{negative}.$

2. For $x \in (-3, -1)$, choose $x = -2$: $(-2 - 9)(-2 + 1)(-2 + 3) = (-11)(-1)(1) = 11 \quad \Rightarrow \quad \text{positive}.$

3. For $x \in (-1, 9)$, choose $x = 0$: $(0 - 9)(0 + 1)(0 + 3) = (-9)(1)(3) = -27 \quad \Rightarrow \quad \text{negative}.$

4. For $x \in (9, \infty)$, choose $x = 10$: $(10 - 9)(10 + 1)(10 + 3) = (1)(11)(13) = 143 \quad \Rightarrow \quad \text{positive}.$

Step 4: Determine where the expression is negative

We need the intervals where the product is negative, as we are solving $(x-9)(x+1)(x+3) < 0$.

From the sign analysis, the expression is negative in the intervals: $(-\infty, -3) \quad \text{and} \quad (-1, 9)$

Step 5: Combine the solution

The solution in interval notation is: $(-\infty, -3) \cup (-1, 9)$

Final Answer:

$\boxed{(-\infty, -3) \cup (-1, 9)}$

Would you like more details or have any questions?

Here are some related questions for further practice:

1. Solve $(x-4)(x+2)(x-7) > 0$.
2. Solve $(x-5)(x+3)(x-2) \geq 0$.
3. Solve $-2(x+1)(x-6)(x-2) > 0$.
4. Solve $(x-8)(x+4)^2(x-1) < 0$.
5. Solve $(x-1)(x+3) < 0$.

Tip: Always check the sign of the expression between critical points to solve polynomial inequalities.