To solve the equation \( x \cdot (x^2 + 3x) = 0 \), you can apply the zero product property, which states that if a product of factors equals zero, at least one of the factors must be zero.

The equation can be broken down into two factors:

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

Now, solve each factor:

### 1. \( x = 0 \)
This is already solved. One solution is \( x = 0 \).

### 2. \( x^2 + 3x = 0 \)
Factor the quadratic expression:

\[
x(x + 3) = 0
\]

This gives us two more potential solutions:

- \( x = 0 \) (which we've already found)
- \( x + 3 = 0 \)

Solve \( x + 3 = 0 \):

\[
x = -3
\]

### Final Answer:
The solutions to the equation are:

\[
x = 0 \quad \text{or} \quad x = -3
\]

Would you like further details on any of these steps, or do you have any questions?

Here are 5 related questions for further exploration:

1. How would you solve the equation \( x^2 + 3x = 4 \)?
2. What is the geometric interpretation of the solutions \( x = 0 \) and \( x = -3 \) on a graph?
3. How can you solve \( (x - 2)(x + 5) = 0 \) using the zero product property?
4. What are the roots of the quadratic equation \( x^2 + 3x + 2 = 0 \)?
5. How does factoring help in solving quadratic equations?

**Tip:** Always check each solution by substituting it back into the original equation to ensure it satisfies the equation.

Learn how to solve the equation x * (x^2 + 3x) = 0 with a detailed step-by-step solution. Understand the application of the zero product property and solve both linear and quadratic equations. Suitable for high school students and those interested in algebraic equations.

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