Factor $$ -15 y^{2}+7 y+2 $$ completely

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UpStudy AI · Accepted Answer

To factor the quadratic expression $$ -15 y^{2} + 7 y + 2 $$ completely, follow these steps:

1. **Factor Out the Negative Sign:**
   
   It's often easier to work with positive leading coefficients. Factor out a $$-1$$:
   $$
   -15 y^{2} + 7 y + 2 = - (15 y^{2} - 7 y - 2)
   $$

2. **Factor the Quadratic Inside the Parentheses:**
   
   Now, focus on factoring $$15 y^{2} - 7 y - 2$$. We need to find two numbers that multiply to $$15 \times (-2) = -30$$ and add up to $$-7$$.

The numbers $$-10$$ and $$3$$ satisfy these conditions:
   $$
   -10 \times 3 = -30 \quad \text{and} \quad -10 + 3 = -7
   $$

3. **Split the Middle Term and Factor by Grouping:**
   
   Rewrite the quadratic using the numbers found:
   $$
   15 y^{2} - 10 y + 3 y - 2
   $$
   
   Group the terms:
   $$
   (15 y^{2} - 10 y) + (3 y - 2)
   $$
   
   Factor out the greatest common factor from each group:
   $$
   5 y (3 y - 2) + 1 (3 y - 2)
   $$
   
   Notice that $$(3 y - 2)$$ is a common factor:
   $$
   (5 y + 1)(3 y - 2)
   $$

4. **Combine with the Factored-Out Negative Sign:**
   
   Don't forget the $$-1$$ we factored out initially:
   $$
   - (5 y + 1)(3 y - 2)
   $$

**Final Factored Form:**
$$
- (5 y + 1)(3 y - 2)
$$

**Answer:**
$$
-\, (5y + 1)\, \left(3y - 2\,\right)
$$
$$ -\, (5y + 1)\, \left(3y - 2\,\right) $$

Factor completely

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