To solve the equation \( x^3 + 24x^2 + 44x = 0 \), let's start by factoring the expression.

Given:

\[
x^3 + 24x^2 + 44x = 0
\]

### Step 1: Factor out the common term \( x \).

\[
x(x^2 + 24x + 44) = 0
\]

Now, we have two factors:

1. \( x = 0 \)
2. \( x^2 + 24x + 44 = 0 \)

### Step 2: Solve the quadratic equation \( x^2 + 24x + 44 = 0 \).

We can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For \( ax^2 + bx + c = 0 \), the coefficients are:

- \( a = 1 \)
- \( b = 24 \)
- \( c = 44 \)

Substituting these into the formula:

\[
x = \frac{-24 \pm \sqrt{24^2 - 4(1)(44)}}{2(1)}
\]

\[
x = \frac{-24 \pm \sqrt{576 - 176}}{2}
\]

\[
x = \frac{-24 \pm \sqrt{400}}{2}
\]

\[
x = \frac{-24 \pm 20}{2}
\]

### Step 3: Find the roots.

1. \( x = \frac{-24 + 20}{2} = \frac{-4}{2} = -2 \)
2. \( x = \frac{-24 - 20}{2} = \frac{-44}{2} = -22 \)

### Final Solutions:

The roots of the equation are:

\[
x = 0, \, x = -2, \, \text{and} \, x = -22
\]

### Summary:

The solutions to the equation \( x^3 + 24x^2 + 44x = 0 \) are \( x = 0 \), \( x = -2 \), and \( x = -22 \).

---

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you factor a cubic equation with more complex coefficients?
2. What is the significance of the discriminant in the quadratic formula?
3. How can you solve a cubic equation using the Rational Root Theorem?
4. What are some graphical interpretations of the roots of polynomial equations?
5. How can you verify solutions to a polynomial equation by substitution?

**Tip:** Always check for a common factor first when solving polynomial equations!

Learn how to solve the cubic equation x^3 + 24x^2 + 44x = 0 with a detailed step-by-step solution. This problem covers key algebraic concepts including factoring and the quadratic formula. Suitable for advanced high school students.

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