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The given equation is \( x^3 + 5x^2 + 6x = 0 \).

To solve this, follow these steps:

### Step 1: Factor the equation.
The equation can be factored by factoring out \( x \):
\[
x(x^2 + 5x + 6) = 0
\]
Now, factor the quadratic \( x^2 + 5x + 6 \):
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]
So, the factored form is:
\[
x(x + 2)(x + 3) = 0
\]

### Step 2: Solve for \( x \).
Set each factor equal to zero:
1. \( x = 0 \)
2. \( x + 2 = 0 \Rightarrow x = -2 \)
3. \( x + 3 = 0 \Rightarrow x = -3 \)

### Step 3: Order the solutions.
The solutions are \( x = -3, -2, 0 \).

Would you like more details on any step or have questions? 

Here are 5 related questions to consider:
1. How do you factor quadratic expressions?
2. What is the significance of factoring in solving polynomial equations?
3. Can all cubic equations be factored as easily as this one?
4. How do we know when to factor out common terms in an equation?
5. How does factoring compare to using the quadratic formula?

**Tip:** Always check if you can factor out a common term before solving an equation—it simplifies the process significantly.

Learn how to solve the cubic polynomial equation x^3 + 5x^2 + 6x = 0 with a detailed step-by-step solution. This problem involves factoring, the zero product property, and solving quadratic equations. Suitable for students in Grades 9-12.

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