Let S be the set of all real numbers a such that the function (x^2+5x+a)/(x^2-10x-24) can be expressed as the quotient of two linear functions. What is the sum of the elements of S?
Note that x^2 -10x - 24 = (x - 12) (x + 2)
So we need to have that a = 6 or x = -204
If a = 6 we have that x^2 + 5x + 6 = (x + 2) (x + 3) and our quotient is (x + 3) / ( x -12)
If a = -204 we have that x^2 + 5x - 204 = ( x -12) (x + 17) and our quotient is ( x +17) / (x + 2)
So
S = { 6 , -204 }
And their sum = -198