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The problem to solve is to **factor** the quadratic expression \( 6x^2 + x - 2 \). Let's solve this by following the steps for factoring a trinomial of this form.

The expression is:
\[
6x^2 + x - 2
\]

### Step 1: Identify the coefficients
For the quadratic expression \( ax^2 + bx + c \):
- \( a = 6 \)
- \( b = 1 \)
- \( c = -2 \)

### Step 2: Multiply \( a \) and \( c \)
Multiply the coefficients of \( a \) and \( c \):
\[
6 \times (-2) = -12
\]

### Step 3: Find two numbers that multiply to -12 and add to \( b \) (which is 1)
We need two numbers that multiply to \(-12\) and add to \( 1 \). These numbers are:
\[
4 \text{ and } -3 \quad \text{(since } 4 \times -3 = -12 \text{ and } 4 + (-3) = 1\text{)}
\]

### Step 4: Rewrite the middle term using these two numbers
Now, rewrite the quadratic expression by splitting the middle term \(x\) into two terms using \(4x\) and \(-3x\):
\[
6x^2 + 4x - 3x - 2
\]

### Step 5: Factor by grouping
Group the terms in pairs:
\[
(6x^2 + 4x) - (3x + 2)
\]

Now factor each group:
\[
2x(3x + 2) - 1(3x + 2)
\]

### Step 6: Factor out the common binomial factor
Now factor out the common binomial factor \((3x + 2)\):
\[
(2x - 1)(3x + 2)
\]

### Final Factored Form:
\[
6x^2 + x - 2 = (2x - 1)(3x + 2)
\]

Would you like further clarification or additional details on any of these steps?

Here are five related questions to explore further:
1. How do you solve quadratic equations that can't be factored easily?
2. What is the quadratic formula, and when is it useful?
3. How do you factor a quadratic when the coefficient of \( x^2 \) is negative?
4. Can you explain how to complete the square for quadratic equations?
5. How can you use graphing to confirm the factorization of a quadratic equation?

**Tip:** Always check your factors by expanding them to ensure they return to the original expression.

Factor the quadratic expression 6x^2 + x - 2.

Learn how to factor the quadratic expression 6x^2 + x - 2 with a detailed step-by-step solution. This problem covers key algebraic concepts such as factoring by grouping, and is suitable for students in Grades 9-10.

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