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Could you write it with as few intervals as possible?

To solve the inequality:

\[
-4a(a + 5)^2(4 - a) > 0
\]

we'll follow these steps:

### Step 1: Identify the Roots
1. **Factor each term**:
   - \(-4a\): This term is zero when \(a = 0\).
   - \((a + 5)^2\): This term is zero when \(a = -5\).
   - \((4 - a)\): This term is zero when \(a = 4\).

So, the critical points are \(a = 0\), \(a = -5\), and \(a = 4\).

### Step 2: Test Intervals
We divide the number line into intervals using these critical points: \( (-\infty, -5) \), \( (-5, 0) \), \( (0, 4) \), and \( (4, \infty) \).

### Step 3: Determine the Sign in Each Interval
1. **Interval \((- \infty, -5)\):** 
   - Choose \(a = -6\): The expression becomes \(-4 \cdot (-6) \cdot 1 \cdot (10)\), which is positive.

2. **Interval \((-5, 0)\):** 
   - Choose \(a = -1\): The expression becomes \(-4 \cdot (-1) \cdot 16 \cdot 5\), which is positive.

3. **Interval \((0, 4)\):** 
   - Choose \(a = 2\): The expression becomes \(-4 \cdot 2 \cdot 49 \cdot 2\), which is negative.

4. **Interval \((4, \infty)\):** 
   - Choose \(a = 5\): The expression becomes \(-4 \cdot 5 \cdot 100 \cdot (-1)\), which is positive.

### Step 4: Write the Solution in Interval Notation
The inequality \(-4a(a + 5)^2(4 - a) > 0\) is satisfied when the expression is positive. From the intervals above, this happens in:
\[
(-\infty, -5) \cup (-5, 0) \cup (4, \infty)
\]

### Final Answer
\[
(-\infty, -5) \cup (-5, 0) \cup (4, \infty)
\]

Would you like further details on any step?

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Here are related questions to explore:

1. How would the solution change if the inequality were \(\geq 0\)?
2. What happens if we change the sign on the term \(-4a\)?
3. Can you solve a similar inequality if \((a+5)^2\) was replaced with \((a+5)^3\)?
4. How does the behavior of \((a+5)^2\) influence the intervals of positivity?
5. What if we added a constant to the expression?

**Tip:** When solving inequalities involving factored expressions, testing values within intervals based on critical points can reveal the solution quickly.

Solve the inequality, Write the solution set in interval notation if possible. Simplify your answer, if necessary. -4a(a + 5)^2(4 - a) > 0

Explore how to solve the polynomial inequality -4a(a + 5)^2(4 - a) > 0, including finding critical points, testing intervals, and writing the solution set in interval notation. This step-by-step solution is suitable for high school students.

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