If (x^2+x-12) divides x^3+ax^2+bx-84 completely,then find the values of a and b.

x^2+x-12 = (x-3)(x+4)

According to the given condition,

x^2+x-12 = (x-3)(x+4) is a factor of x^3+ax^2+bx-84 = f(x).

 

Hence, f(3) = f(-4) = 0

Therefore, 

27+9a+3b-84=0  or 3a+b-19=0

-64 + 16a-4b-84=0 or 4a-b-37=0

 

Adding these two equations, we get, 7a - 56 = 0 or a = 8

So, b = 19-24 = -5