It's (x^3-3x^2+9x-29)/8, with remainder 87.
First way to do it is a standard long division algorithm, I guess you know that. In short, you divide x^4 with 8x (getting x^3/8), then multiply that with 8x+24 (getting x^4+3x^3), and subtract that from x^4-2x (getting -3x^3-2x). Then you do the same thing with that result instead of starting numerator, until you get a single number that cannot be divided by 8x - and that number is 87.
Second way is more interesting, and uses Horner's algorithm. It's easier to apply, but harder to see why it works. It works only when denominator is x-alpha for some alpha, so we'll divide by x+3 (alpha being -3), and then divide by 8 separately.
You make a table with 1, 0, 0, -2, 0 (coefficients of numerator) in the first row, and in the second row, drop 1, multiply by alpha(=-3) and add 0 (getting -3, then multiply that by alpha and add next 0 (getting 9), multiply that by alpha and add -2 (getting -29), and multiply that by alpha and add the last 0 (getting 87). Entries except the last in the second row are the coefficients of quotient (remember that they still have to be divided by 8), and the last entry is the remainder.
The whole process is explained at http://en.wikipedia.org/wiki/Synthetic_division.