If (ax+2)(bx+7) = 15x^2 + cx +14 for all values of x, and a+b =8, what are two possible values of c ? a. 3 and 5 b. 10 and 21 c. 31 and 41 d. 6 and 35

For the two quadratics to match, the coefficients of corresponding terms must be equal, so that

From the given options, we are only allowing to be an integer, which means must also be integers. Then we have eight possible choices for :

1, 15

3, 5

5, 3

15, 1

plus the same combinations with negative signs on both numbers. But since the choices for are also positive, we can ignore the negative versions.

If , then .

If , then .

If , then .

If , then .

So the answer is C.