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Let's factor the expression \( 6x^3 - 21x^2 - 12x \).

### Step 1: Factor out the greatest common factor (GCF)
The GCF of the coefficients \(6\), \(21\), and \(12\) is \(3\), and the GCF of the variables is \(x\) (since all terms contain \(x\)).

Thus, the GCF is \(3x\).

### Step 2: Factor the expression using the GCF
Factor out \(3x\) from each term:

\[
6x^3 - 21x^2 - 12x = 3x(2x^2 - 7x - 4)
\]

### Step 3: Factor the quadratic expression
Now, we need to factor the quadratic expression \(2x^2 - 7x - 4\). To do this, we will use the method of splitting the middle term.

We are looking for two numbers that multiply to \(2 \times (-4) = -8\) and add to \(-7\). These numbers are \(-8\) and \(1\).

So, we rewrite the middle term \(-7x\) as \(-8x + x\):

\[
2x^2 - 8x + x - 4
\]

### Step 4: Factor by grouping
Now, group the terms:

\[
(2x^2 - 8x) + (x - 4)
\]

Factor out the common factors from each group:

\[
2x(x - 4) + 1(x - 4)
\]

Now factor out the common binomial factor \((x - 4)\):

\[
(2x + 1)(x - 4)
\]

### Step 5: Combine the factors
So, the fully factored form of the original expression is:

\[
3x(2x + 1)(x - 4)
\]

This is the factored form of \(6x^3 - 21x^2 - 12x\).

Would you like any further details or explanations on this? 

Here are 5 related questions to expand on this topic:
1. How do you identify the greatest common factor in a polynomial?
2. Can all quadratic expressions be factored using integer coefficients?
3. How do you know when to use the method of splitting the middle term?
4. What other methods can you use to factor quadratic expressions?
5. How can you check if your factorization is correct?

**Tip**: Always check your factorization by expanding the factors to ensure it matches the original expression.

Factor 6x^3 - 21x^2 - 12x over the integers.

Learn how to factor the polynomial 6x^3 - 21x^2 - 12x with a detailed, step-by-step solution. This tutorial covers the key concepts of factoring out the greatest common factor (GCF) and factoring quadratic expressions using algebraic methods. Suitable for students in Grades 9-11.

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