Fully factor 12+ 21x - 6x^2 show all steps using the factor method

We are tasked with fully factoring the quadratic expression \( 12 + 21x - 6x^2 \). To factor this expression using the factoring method, we'll follow these steps:

### Step 1: Arrange in standard form
First, arrange the expression in the standard form of a quadratic, \( ax^2 + bx + c \). This expression is given as:
\[
12 + 21x - 6x^2
\]
Rearranging it in the form \( ax^2 + bx + c \), we get:
\[
-6x^2 + 21x + 12
\]

### Step 2: Factor out the greatest common factor (GCF)
The coefficients of all the terms \(-6x^2 + 21x + 12\) share a common factor of 3. Factor out the GCF (which is \(-3\) to keep the leading coefficient positive inside the parentheses):
\[
-3(2x^2 - 7x - 4)
\]

### Step 3: Factor the quadratic trinomial
Now, we need to factor the quadratic expression \( 2x^2 - 7x - 4 \). To do this using the "factoring by grouping" method, we follow these steps:

1. Multiply the coefficient of \(x^2\) (which is 2) by the constant term (which is \(-4\)):
   \[
   2 \times (-4) = -8
   \]
2. Find two numbers that multiply to \(-8\) and add up to the middle term's coefficient, \(-7\). These numbers are \(-8\) and \(1\), because:
   \[
   -8 \times 1 = -8 \quad \text{and} \quad -8 + 1 = -7
   \]
3. Rewrite the middle term \(-7x\) as \(-8x + x\):
   \[
   2x^2 - 8x + x - 4
   \]
4. Group the terms in pairs and factor each pair:
   \[
   (2x^2 - 8x) + (x - 4)
   \]
   Factor out the greatest common factor from each group:
   \[
   2x(x - 4) + 1(x - 4)
   \]
5. Now, factor out the common binomial factor \((x - 4)\):
   \[
   (2x + 1)(x - 4)
   \]

### Step 4: Write the final factored form
Now, substitute back the factor \(-3\) that we factored out in Step 2:
\[
\boxed{-3(2x + 1)(x - 4)}
\]
This is the fully factored form of the quadratic expression.

### Summary of steps:
1. Rewrite the quadratic in standard form.
2. Factor out the greatest common factor \(-3\).
3. Factor the trinomial using factoring by grouping.
4. Combine the factors to get the final factored form.

---

Do you have any questions or would you like more details about any of these steps?

Here are 5 related questions you might find useful:
1. How do you factor quadratics when the leading coefficient isn't 1?
2. What is the greatest common factor (GCF) and how do you find it?
3. How do you recognize when factoring by grouping is the best method?
4. What other methods can be used to factor quadratics?
5. How do you verify that a factored expression is correct?

**Tip**: Always check your factored form by expanding the factors back to the original expression to ensure correctness.

Fully factor 12 + 21x - 6x^2 show all steps using the factor method.

Learn how to fully factor the quadratic expression 12 + 21x - 6x^2 step by step. This solution covers factoring by grouping and using the greatest common factor, essential for high school algebra students in Grades 9-11.

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