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.