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Let's solve the inequality \(-4(x + 8)(x + 7)(x + 3) < 0\).

### Step 1: Simplify the inequality
Since \(-4\) is a constant negative number, we can divide both sides of the inequality by \(-4\), but remember that dividing by a negative number reverses the inequality sign.

\[
(x + 8)(x + 7)(x + 3) > 0
\]

Now we need to solve \((x + 8)(x + 7)(x + 3) > 0\).

### Step 2: Find the roots of the polynomial
The expression is already factored, and the roots (values of \(x\) where the product equals 0) are:
- \(x = -8\)
- \(x = -7\)
- \(x = -3\)

These roots divide the number line into four intervals:
- \( (-\infty, -8) \)
- \( (-8, -7) \)
- \( (-7, -3) \)
- \( (-3, \infty) \)

### Step 3: Test the sign of the expression in each interval
We will test the sign of \((x + 8)(x + 7)(x + 3)\) in each interval by picking test points within these intervals.

1. **For \(x < -8\)** (choose \(x = -9\)):
   \[
   (x + 8)(x + 7)(x + 3) = (-9 + 8)(-9 + 7)(-9 + 3) = (-1)(-2)(-6) = -12 \quad (\text{negative})
   \]
   
2. **For \(-8 < x < -7\)** (choose \(x = -7.5\)):
   \[
   (-7.5 + 8)(-7.5 + 7)(-7.5 + 3) = (0.5)(-0.5)(-4.5) = 1.125 \quad (\text{positive})
   \]

3. **For \(-7 < x < -3\)** (choose \(x = -5\)):
   \[
   (-5 + 8)(-5 + 7)(-5 + 3) = (3)(2)(-2) = -12 \quad (\text{negative})
   \]

4. **For \(x > -3\)** (choose \(x = 0\)):
   \[
   (0 + 8)(0 + 7)(0 + 3) = (8)(7)(3) = 168 \quad (\text{positive})
   \]

### Step 4: Determine the solution
We want the expression to be positive, so we select the intervals where the sign is positive.

The solution is:
\[
(-8, -7) \cup (-3, \infty)
\]

### Final Answer:
\[
x \in (-8, -7) \cup (-3, \infty)
\]

Would you like more details on any part of the solution, or do you have any questions?

Here are five questions that can expand on this:
1. How does the constant factor \(-4\) influence the inequality?
2. What happens if the inequality sign is changed to \(\leq\) instead of \(<\)?
3. Can this solution be visualized graphically?
4. How would you handle this inequality if the polynomial were not factored?
5. What are other methods to solve polynomial inequalities?

**Tip**: Always remember to reverse the inequality sign when dividing or multiplying by a negative number.

Solve the polynomial inequality: -4(x + 8)(x + 7)(x + 3) < 0.

Learn how to solve the polynomial inequality -4(x + 8)(x + 7)(x + 3) < 0 with a detailed step-by-step solution. This problem involves algebraic concepts, including polynomial inequalities and sign analysis. Suitable for students in Grades 10-12.

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