Solve the inequality. (Enter your answer using interval notation.)
(x + 9)2(x + 3)(x − 1) > 0

We are tasked with solving the inequality:

\[
(x + 9)^2(x + 3)(x - 1) > 0
\]

### Step 1: Identify critical points
The critical points are the values of \(x\) where the expression equals zero. To find these points, set each factor in the expression to zero:

1. \((x + 9)^2 = 0 \Rightarrow x = -9\)
2. \((x + 3) = 0 \Rightarrow x = -3\)
3. \((x - 1) = 0 \Rightarrow x = 1\)

Thus, the critical points are \(x = -9\), \(x = -3\), and \(x = 1\).

### Step 2: Analyze the intervals
The critical points divide the real number line into four intervals:

1. \( (-\infty, -9) \)
2. \( (-9, -3) \)
3. \( (-3, 1) \)
4. \( (1, \infty) \)

We now analyze the sign of the expression \((x + 9)^2(x + 3)(x - 1)\) on each interval.

#### Interval 1: \( (-\infty, -9) \)
- Choose \(x = -10\):
  - \((x + 9)^2 = (-10 + 9)^2 = 1\) (positive)
  - \((x + 3) = (-10 + 3) = -7\) (negative)
  - \((x - 1) = (-10 - 1) = -11\) (negative)

  The product is positive: \( (+)(-)(-) = + \).

#### Interval 2: \( (-9, -3) \)
- Choose \(x = -5\):
  - \((x + 9)^2 = (-5 + 9)^2 = 16\) (positive)
  - \((x + 3) = (-5 + 3) = -2\) (negative)
  - \((x - 1) = (-5 - 1) = -6\) (negative)

  The product is positive: \( (+)(-)(-) = + \).

#### Interval 3: \( (-3, 1) \)
- Choose \(x = 0\):
  - \((x + 9)^2 = (0 + 9)^2 = 81\) (positive)
  - \((x + 3) = (0 + 3) = 3\) (positive)
  - \((x - 1) = (0 - 1) = -1\) (negative)

  The product is negative: \( (+)(+)(-) = - \).

#### Interval 4: \( (1, \infty) \)
- Choose \(x = 2\):
  - \((x + 9)^2 = (2 + 9)^2 = 121\) (positive)
  - \((x + 3) = (2 + 3) = 5\) (positive)
  - \((x - 1) = (2 - 1) = 1\) (positive)

  The product is positive: \( (+)(+)(+) = + \).

### Step 3: Consider the critical points
- At \(x = -9\), \((x + 9)^2 = 0\), so the expression is zero.
- At \(x = -3\) and \(x = 1\), the expression is zero because of the \((x + 3)\) and \((x - 1)\) factors, respectively.

### Step 4: Solution in interval notation
The expression is positive on the intervals where the product is positive. These intervals are:
- \( (-\infty, -9) \)
- \( (-9, -3) \)
- \( (1, \infty) \)

Since we want \( (x + 9)^2(x + 3)(x - 1) > 0 \), we exclude the points where the expression equals zero.

Thus, the solution is:

\[
(-\infty, -9) \cup (-9, -3) \cup (1, \infty)
\]

Would you like any more details or clarifications?

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Here are 5 related questions:
1. How would the solution change if the inequality were \( \geq 0 \)?
2. What happens if you replace one of the linear factors with a higher-degree term?
3. How do we handle inequalities with rational expressions?
4. What is the role of multiplicity in determining the sign of intervals?
5. How does squaring a factor like \((x + 9)\) affect the sign analysis?

**Tip:** When solving inequalities, always remember to test values from each interval to check the sign of the expression.

Solve the inequality. (Enter your answer using interval notation.) (x + 9)^2(x + 3)(x − 1) > 0

Learn how to solve the inequality (x + 9)^2(x + 3)(x − 1) > 0 with a step-by-step guide. This problem covers polynomial inequalities, sign analysis, and critical points. Suitable for students in Grades 10-12.

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