To factor the trinomial \(12y^2 + 28y - 5\), you can use the quadratic factoring method. Here's how you can do it:
1. Multiply the coefficient of the leading term (12) by the constant term (-5). This gives you -60.
2. Find two numbers that multiply to -60 and add up to the coefficient of the middle term (28). These numbers are 30 and -2 because \(30 \times (-2) = -60\) and \(30 + (-2) = 28\).
3. Rewrite the middle term (28y) using these two numbers:
\(12y^2 + 30y - 2y - 5\)
4. Now, group the terms and factor by grouping:
\( (12y^2 + 30y) - (2y + 5) \)
5. Factor out the greatest common factor from each group:
\(6y(2y + 5) - 1(2y + 5)\)
6. Notice that you now have a common factor of \(2y + 5\):
\( (2y + 5)(6y - 1) \)
So, \(12y^2 + 28y - 5\) factors into \((2y + 5)(6y - 1)\).