here you are given x^3/( 3 + 5x) = x^3 * ( 1/(3 + 5x) )
Here we observe that the only component that truely requires a taylor expansion is (1/(3+ 5x)), whose derivatives are far easier to compute than that of the original function.
Once this is obtained you simply multiple the series through by x^3 and viola you have answered the problem with only a 10th of the pain!!
Hope this points you in the right direction,
Let me know if you require any further assistance,
ps - make life even easier on yourself by simplifying the function even further, i.e.
1/(3 + 5x) = 1/( 3 (1 + ((5/3)x)) = (1/3)(1/(1 + (5/3)x)) i.e bring the 1/3 out the front
furthermore let t = 5/3x and thus you you need only solve for 1/(1+t) = a0 + a1*t + ....
To then get in back in terms of x, sub t = 5/3x into the the taylor expansion and your done!