The answer is choice B
2(3n+7)(2n-7)
--------------------------------------------------------
There are two ways to do this
Method 1: Factor by grouping
First pull out the GCF 2
12n^2 - 14n - 98
2(6n^2 - 7n - 49)
Then we need to find two numbers that multiply to 6*(-49) = -294 and that add to -7 which is the middle coefficient.
Those two numbers are -21 and 14
-21 * 14 = -294
-21 + 14 = -7
So we break down the -7n into -21n + 14n and factor by grouping
2(6n^2 - 7n - 49)
2*(6n^2 - 21n + 14n - 49)
2*((6n^2-21n) + (14n-49))
2*(3n(2n - 7) + 7(2n - 7))
2(3n + 7)(2n - 7)
-----------------------------
Method 2: The quadratic formula
First pull out the GCF 2
12n^2 - 14n - 98
2(6n^2 - 7n - 49)
Now solve 6n^2 - 7n - 49 = 0 for n
Use the quadratic formula to do so
Recall that the quadratic formula is
x = (-b+-sqrt(b^2-4ac))/(2*a)
So in this case, we have
n = (-b+-sqrt(b^2-4ac))/(2*a)
n = (-(-7)+-sqrt((-7)^2-4*6*(-49)))/(2*6)
n = (7+-sqrt(1225))/(12)
n = (7+-35)/(12)
n = (7+35)/(12) or n = (7-35)/(12)
n = 42/12 or n = -28/12
n = 7/2 or n = -7/3
Now use the two solutions n = 7/2 and n = -7/3 to find the factorization for 6n^2 - 7n - 49
n = 7/2 and n = -7/3
2n = 7 and 3n = -7
2n-7 = 0 and 3n+7 = 0
(2n - 7)(3n + 7) = 0
(3n + 7)(2n - 7) = 0
So that shows 6n^2 - 7n - 49 factors to (3n + 7)(2n - 7)
Therefore, 12n^2 - 14n - 98 factors to 2(3n + 7)(2n - 7)
Which is why the answer is choice B
