I think you have the right idea, ND....let's look at the factoring once more...
2x^2 - 5x - 7 = (2x -7) (x + 1)
So the roots are 7/2 and -1
So
(7/2)/(-1) + (-1)/(7/2) =
-7/2 - 2/7 = -(7/2 + 2/7) = - (49 + 4)/14 = -53/14
I think that might be it ......