I'm not sure I understand your question. Your calculations are correct. Just solve the remaining integral .
So I am solving integral sign (S) (x-1)e^-2xdx by using integration by parts
u = x-1 dv = e^-2x dx
du = dx v= (-1/2)e^-2x
so S udv = uv - S vdu
so = (x-1)((-1/2)e^-2x) - S (-1/2)e^-2x dx
Now my question is when I solve the S dx part does it become x-1 or just x?