Okay, let's tackle this problem step by step. The question is about finding the sum of the vertical asymptotes and the horizontal asymptote of the function f(x) = (2x)/(x² - 5x - 14). So, we need to find a, b, and c where x = a and x = b are the vertical asymptotes and y = c is the horizontal asymptote. Then, we add them all together to get a + b + c.

First, I remember that vertical asymptotes occur where the denominator of a rational function is zero, provided that the numerator isn't also zero at those points. So, I need to factor the denominator and find its roots. The denominator here is x² - 5x - 14. Let me try to factor that. 

Looking for two numbers that multiply to -14 and add up to -5. Hmm, factors of -14 could be -7 and 2 because (-7) * 2 = -14 and (-7) + 2 = -5. Yeah, that works. So, the denominator factors into (x - 7)(x + 2). Therefore, the vertical asymptotes are at x = 7 and x = -2. So, a = 7 and b = -2. 

Wait, let me double-check that. If I expand (x - 7)(x + 2), it should be x² + 2x - 7x -14 = x² -5x -14. Yep, that's correct. So, the vertical asymptotes are indeed at 7 and -2. 

Next, the horizontal asymptote. For rational functions, the horizontal asymptote depends on the degrees of the numerator and denominator. The numerator here is 2x, which is degree 1, and the denominator is x² -5x -14, which is degree 2. Since the degree of the denominator is higher than the degree of the numerator, the horizontal asymptote is y = 0. 

Wait, let me confirm. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If they are equal, it's the ratio of the leading coefficients. If the numerator is higher, there's no horizontal asymptote, but there might be an oblique one. Here, since denominator is higher, y = 0 is the horizontal asymptote. So, c = 0. 

Therefore, a = 7, b = -2, c = 0. Adding them together: 7 + (-2) + 0 = 5. 

But let me check again to make sure I didn't make any mistakes. For the vertical asymptotes, the denominator factors correctly. Yes, (x -7)(x +2) gives x² -5x -14. For the horizontal asymptote, since the denominator is degree 2 and the numerator is degree 1, it's indeed y = 0. So, adding 7, -2, and 0 gives 5. 

I think that's all. The answer should be 5. 

Wait, just to be thorough, let me check the horizontal asymptote again. For rational functions, when the degree of the numerator is less than the denominator, horizontal asymptote is y = 0. If they are equal, it's the ratio of the leading coefficients. If the numerator is higher, there's no horizontal asymptote. So, in this case, since denominator is higher, horizontal asymptote is y = 0. Correct. 

And for vertical asymptotes, we can test the function near x =7 and x = -2. For example, approaching 7 from the left and right, the denominator approaches 0, and the function goes to positive or negative infinity. Same with x = -2. So, those are indeed vertical asymptotes. 

Therefore, the sum is 7 + (-2) + 0 = 5. I think that's the answer.
Therefore, the sum of the vertical asymptotes and the horizontal asymptote is $\boxed{5}$.
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