To factor the expression \( 5x^2 - 3x - 2 \), we will proceed step by step:

### Step 1: Identify the coefficients
The given expression is a quadratic in the form of \( ax^2 + bx + c \), where:
- \( a = 5 \)
- \( b = -3 \)
- \( c = -2 \)

### Step 2: Multiply \( a \) and \( c \)
We need to multiply the coefficient of \( x^2 \) (which is 5) by the constant term (which is -2):
\[
a \times c = 5 \times (-2) = -10
\]

### Step 3: Find two numbers that multiply to -10 and add to -3
We need to find two numbers that multiply to -10 and add to -3. These numbers are:
- \( -5 \) and \( 2 \), because:
  \[
  (-5) \times 2 = -10 \quad \text{and} \quad (-5) + 2 = -3
  \]

### Step 4: Rewrite the middle term using these two numbers
Rewrite \( -3x \) as \( -5x + 2x \):
\[
5x^2 - 5x + 2x - 2
\]

### Step 5: Factor by grouping
Now, group the terms:
\[
(5x^2 - 5x) + (2x - 2)
\]
Factor out the greatest common factor (GCF) from each group:
\[
5x(x - 1) + 2(x - 1)
\]

### Step 6: Factor out the common binomial
Now, factor out \( (x - 1) \):
\[
(5x + 2)(x - 1)
\]

### Final Factored Form:
\[
5x^2 - 3x - 2 = (5x + 2)(x - 1)
\]

Would you like further explanation or have any questions on this process?

Here are 5 related questions you might find interesting:
1. How do you solve a quadratic equation using factoring?
2. What is the quadratic formula and how is it derived?
3. How do you complete the square to solve a quadratic equation?
4. Can you solve quadratic equations that don't factor neatly?
5. What are the special cases of factoring quadratic expressions?

**Tip**: When factoring quadratics, always start by looking for a pair of numbers that multiply to \( ac \) and add to \( b \).

Learn how to factor the quadratic expression 5x^2 - 3x - 2 step by step. This solution includes identifying coefficients, multiplying 'a' and 'c', and factoring by grouping. Suitable for students in Grades 9-10.

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