Factor completely. $$ 3 w^{2}-5 w-2 $$ Select the correct choice and fill in any answer boxes within your choice. A. $$ 3 w^{2}-5 w-2=\square $$ B. The expression is prime.

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To factor the quadratic expression $$ 3w^{2} - 5w - 2 $$ completely, follow these steps:

1. **Identify the coefficients:**
   - $$ a = 3 $$ (coefficient of $$ w^2 $$)
   - $$ b = -5 $$ (coefficient of $$ w $$)
   - $$ c = -2 $$ (constant term)

2. **Find two numbers that multiply to $$ a \times c = 3 \times (-2) = -6 $$ and add up to $$ b = -5 $$:**
   - The numbers are **-6** and **1** because:
     - $$ -6 \times 1 = -6 $$
     - $$ -6 + 1 = -5 $$

3. **Rewrite the middle term using these numbers:**
   $$
   3w^{2} - 6w + w - 2
   $$

4. **Factor by grouping:**
   $$
   (3w^{2} - 6w) + (w - 2) = 3w(w - 2) + 1(w - 2)
   $$

5. **Factor out the common binomial factor $$ (w - 2) $$:**
   $$
   (3w + 1)(w - 2)
   $$

**Final Factored Form:**

**A.** $$ 3w^{2} - 5w - 2 = (3w + 1)(w - 2) $$
$$ 3w^{2} - 5w - 2 = (3w + 1)(w - 2) $$

Factor completely. \( 3 w^{2}-5 w-2 \) Select the correct choice and fill in any answer boxes within your choice. A. \( 3 w^{2}-5 w-2=\square \) B. The expression is prime.

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