Asymptotes Question

As follows:

$f(x) = \frac{2x-8}{x^2 -2x - 3} \qquad\text{ and }\qquad g(x) = \frac{3x+9}{2x-4}$

I found    $f( \frac{3x+9}{2x-4})$       and then I simplified it all down.

There was a lot of algebra and I have not checked my result.  Checking could be done by graphing the original substituted function

anyway I ended up with    [ -40(x^2+....)]  /  [-15(x^2+....) ]   if this is correct I would assume that the horizontal asymptote is   

40/15 = 8/3

But there could be 100 mistakes in my working and also in my logic.