The expression you provided is:

\[
12x^3 + 18x^2 - 24x
\]

This can be factored by first identifying the greatest common factor (GCF) of the terms.

### Step 1: Find the GCF
The GCF of \(12x^3\), \(18x^2\), and \(-24x\) is \(6x\), because:
- The GCF of the coefficients (12, 18, -24) is 6.
- The lowest power of \(x\) among the terms is \(x\), so we factor out \(x\).

### Step 2: Factor out the GCF
Now, factor out \(6x\) from the entire expression:

\[
12x^3 + 18x^2 - 24x = 6x(2x^2 + 3x - 4)
\]

### Step 3: Factor the quadratic (if possible)
Next, we will attempt to factor the quadratic \(2x^2 + 3x - 4\).

To factor \(2x^2 + 3x - 4\), we look for two numbers that multiply to \(2 \times -4 = -8\) and add up to \(3\). The numbers are 4 and -2.

So, rewrite the middle term:

\[
2x^2 + 4x - 2x - 4
\]

Group the terms:

\[
(2x^2 + 4x) - (2x + 4)
\]

Factor out the common factors from each group:

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

Now factor out \((x + 2)\):

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

Factor out the common factor of 2 from the first binomial:

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

### Final Factored Form:
Now, substituting this back into the original expression:

\[
12x^3 + 18x^2 - 24x = 6x \cdot 2(x - 1)(x + 2)
\]

This simplifies to:

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

So, the factored form of \(12x^3 + 18x^2 - 24x\) is:

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

Would you like more details or have any questions?

### Related questions:
1. How do you find the greatest common factor (GCF) of a polynomial?
2. How do you factor quadratic expressions of the form \(ax^2 + bx + c\)?
3. Can you explain the grouping method in more detail for factoring?
4. What are some alternative methods to factor polynomials?
5. How do you solve for the roots of the factored expression \(12x(x - 1)(x + 2)\)?

### Tip:
Always check if you can factor out the greatest common factor (GCF) before attempting more complex factoring methods.

Learn how to factor the expression 12x^3 + 18x^2 - 24x using the greatest common factor and factoring a quadratic expression. This step-by-step solution covers key algebraic concepts and techniques such as GCF, quadratic factoring, and the grouping method. Suitable for high school students in Grades 9-12.

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