I took the derivative of function, yielding the result: $$\frac{-3x^4 - 4x^3 - 3x^2 + 18x + 6}{x^6 - 6x^3 + 9}$$ I know this to be correct and wanted to simplify the answer.

I've never attempted long division on an expression in which the highest order is in the denominator before but attempted:

$$
\require{enclose}
\begin{array}{r11}
 -3x^{-2} - 4x^{-3} - 3x^{-4} ... \\[-3pt]
 x^6 - 6x^3 + 9 \enclose{longdiv}{-3x^4 - 4x^3 - 3x^2 + 18x + 6}\kern-.2ex \\[-3pt]
 \underline{-3x^4-18x+27x^{-2} } \\[-3pt]
 -4x^3-3x^2+36x+6 \\[-3pt]
 \underline{-4x^3-24-36x^{-3}} \\[-3pt]
 -3x^2+36x-18-+27x^{-2}-36x^{-3} \\[-3pt]
 ...
 \end{array}
$$

It seems that for every term I eliminate, I gain another of a lower order. Is there a problem with my method, do these two expressions not divide cleanly? 

How can one tell whether the division of polynomials is possible?