Since the given sequence is a geometric progression, the ratio   is the same as the ratio 

and both are equal to the common ratio.  It gives you an equation


 = .


To solve it, cross multiply


(2x-15)*(2x-15) = (x-12)*(3x+30),  ====>

4x^2 - 30x + 225 = 3x^2 -36x + 30x - 360  ====>

x^2 - 24x + 585 = 0.


Do not hurry to solve this quadratic equation.
You do not need do it.
They ask you about the sum of all possible solutions.


Apply the Vieta's theorem, which says that the sum of all roots of this equation is equal to the coefficient at "x" taken with the opposite sign.


Therefore, you answer is  24.