To solve the inequality $x^3(x - 3)^2(x + 5) < 0$, let’s analyze the factors and their behavior over different intervals.

Step 1: Identify the Roots

The inequality can be broken down by analyzing each factor separately:

1. $x^3 = 0$ implies $x = 0$.
2. $(x - 3)^2 = 0$ implies $x = 3$.
3. $(x + 5) = 0$ implies $x = -5$.

These roots divide the real number line into four intervals:

• $(-\infty, -5)$
• $(-5, 0)$
• $(0, 3)$
• $(3, \infty)$

Step 2: Determine the Sign of Each Factor in Each Interval

For the inequality to hold, the product of all factors must be less than zero (negative). Let’s analyze the sign of each factor in each interval:

1. Interval $(-\infty, -5)$:

   • $x^3$ is negative.
   • $(x - 3)^2$ is positive (since it’s squared, it’s always positive).
   • $(x + 5)$ is negative.
   • Product: Negative $\times$ Positive $\times$ Negative = Positive.

2. Interval $(-5, 0)$:

   • $x^3$ is negative.
   • $(x - 3)^2$ is positive.
   • $(x + 5)$ is positive.
   • Product: Negative $\times$ Positive $\times$ Positive = Negative.

3. Interval $(0, 3)$:

   • $x^3$ is positive.
   • $(x - 3)^2$ is positive.
   • $(x + 5)$ is positive.
   • Product: Positive $\times$ Positive $\times$ Positive = Positive.

4. Interval $(3, \infty)$:

   • $x^3$ is positive.
   • $(x - 3)^2$ is positive.
   • $(x + 5)$ is positive.
   • Product: Positive $\times$ Positive $\times$ Positive = Positive.

Step 3: Identify Where the Product is Negative

From the analysis, the product is negative only in the interval $(-5, 0)$.

Step 4: Express the Solution in Interval Notation

Since the inequality is strictly less than zero, we do not include the endpoints where the product equals zero. Thus, the solution is:

$(-5, 0)$

Let me know if you’d like further details or have any questions.

Here are some related questions to deepen understanding:

1. How would the solution change if the inequality were $\leq 0$ instead of $< 0$?
2. What happens if we change the exponent on $(x - 3)^2$ to an odd power?
3. Can you solve $x^3(x - 3)^2(x + 5) > 0$ using the same steps?
4. What effect does each factor have on the overall sign of the product in each interval?
5. How would the solution change if another factor, like $(x - 2)$, were added to the inequality?

Tip: Always examine the multiplicity of each root, as it affects whether the sign changes at that root or not.