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The given expression is:

\[
2x^3 + 22x^2 - 24x
\]

### Step 1: Factor out the greatest common factor (GCF)
The GCF of the coefficients \(2\), \(22\), and \(-24\) is \(2\). Additionally, all terms contain \(x\), so the GCF is \(2x\).

\[
2x(x^2 + 11x - 12)
\]

### Step 2: Factor the quadratic expression \(x^2 + 11x - 12\)
Now we need to factor the quadratic expression inside the parentheses: \(x^2 + 11x - 12\).

We are looking for two numbers that multiply to \(-12\) (the constant term) and add up to \(11\) (the coefficient of the middle term).

The two numbers are \(12\) and \(-1\), because:

\[
12 \times (-1) = -12 \quad \text{and} \quad 12 + (-1) = 11
\]

### Step 3: Rewrite and factor
We can now rewrite the quadratic as:

\[
x^2 + 12x - x - 12
\]

Group the terms:

\[
(x^2 + 12x) - (x + 12)
\]

Factor by grouping:

\[
x(x + 12) - 1(x + 12)
\]

Now factor out the common factor \((x + 12)\):

\[
(x - 1)(x + 12)
\]

### Step 4: Final factored form
Now substitute this factored quadratic back into the original expression:

\[
2x(x - 1)(x + 12)
\]

Thus, the completely factored form of the expression is:

\[
2x(x - 1)(x + 12)
\]

Let me know if you need further details or explanations!

### Relative Questions:
1. What are the steps for factoring a quadratic equation?
2. How do you determine the greatest common factor (GCF) of an expression?
3. What is factoring by grouping and when do you use it?
4. Can all cubic polynomials be factored this way?
5. What other methods can be used to factor complex expressions?

### Tip:
Always start by factoring out the greatest common factor (GCF) before attempting more complex factoring techniques!

Learn how to factor the cubic polynomial 2x^3 + 22x^2 - 24x completely using the greatest common factor and factoring quadratic expressions. Detailed step-by-step guide suitable for students in Grades 9-12.

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