factor each trinomial completely and explain how you got it
6x^2 + 7x - 49
-6a^2 - 25a - 25
16b^2 + 60b - 100
For the first trinomial, 6x^2 + 7x - 49, we need to find two numbers that multiply to -294 (6*(-49)) and add up to 7. The numbers are 21 and -14. So we can rewrite the middle term as 21x - 14x and factor by grouping:
6x^2 + 21x - 14x - 49
3x(2x + 7) - 7(2x + 7)
(3x - 7)(2x + 7)
For the second trinomial, -6a^2 - 25a - 25, we need to find two numbers that multiply to -150 (-6*(-25)) and add up to -25. The numbers are -30 and 5. So we can rewrite the middle term as -30a + 5a and factor by grouping:
-6a^2 - 30a + 5a - 25
-6a(a + 5) + 5(a + 5)
(-6a + 5)(a + 5)
For the third trinomial, 16b^2 + 60b - 100, we first need to simplify by dividing through by 4:
4(4b^2 + 15b - 25)
Next, we need to find two numbers that multiply to -100 (4*(-25)) and add up to 15. The numbers are 20 and -5. So we can rewrite the middle term as 20b - 5b and factor by grouping:
4(4b^2 + 20b - 5b - 25)
4((4b^2 + 20b) + (-5b - 25))
4(4b(b + 5) - 5(b + 5))
4(4b - 5)(b + 5)