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To factor the quadratic expression \( 2x^2 + 3x - 2 \), let's go through these steps:

### Step 1: Multiply \( a \) and \( c \)
In this expression, \( a = 2 \), \( b = 3 \), and \( c = -2 \).
We need to find two numbers that multiply to \( a \cdot c = 2 \cdot (-2) = -4 \) and add up to \( b = 3 \).

### Step 2: Find the Numbers
The numbers that multiply to \(-4\) and add to \(3\) are \(4\) and \(-1\) (since \( 4 \cdot (-1) = -4 \) and \( 4 + (-1) = 3 \)).

### Step 3: Rewrite the Middle Term
Rewrite the expression by splitting the middle term \(3x\) into \(4x - x\):
\[
2x^2 + 4x - x - 2
\]

### Step 4: Factor by Grouping
Now, factor out the common terms from each pair:
\[
= 2x(x + 2) - 1(x + 2)
\]

### Step 5: Factor Out the Common Binomial
Now, factor out \((x + 2)\):
\[
= (2x - 1)(x + 2)
\]

### Final Answer
The factored form of \( 2x^2 + 3x - 2 \) is:
\[
(2x - 1)(x + 2)
\]

Would you like further clarification on any steps?

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**Related Questions:**
1. How do we check if the factors are correct?
2. Can all quadratic expressions be factored this way?
3. What if there were no integers that multiply to \(ac\) and add to \(b\)?
4. How can this method be extended to more complex expressions?
5. Why do we factor quadratic expressions in general?

**Tip:** When factoring quadratics, always start by looking for two numbers that multiply to \(ac\) and add to \(b\). This can often simplify the problem significantly.

Factor this quadratic expression: 2x^2 + 3x - 2

Learn how to factor the quadratic expression 2x^2 + 3x - 2 using the ac-method and grouping method. This step-by-step guide helps students understand essential algebraic concepts, suitable for Grades 8-10.

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