I just have to write out the form of the partial fraction decomposition.

$\displaystyle \frac {x^4} {(x^3+x)(x^2-x+3)}$

Now when the fraction is improper you are supposed to use long division right? But how does that work with more than one term in the denominator? I came up with $\displaystyle \frac {A}{x} + \frac {Bx+C}{(x^2+1)} + \frac {Dx+E}{(x^2-x+3)}$ but I don't think that is right as I did not use long division.

The reason I didn't use long division was because if you multiply out the denominator, the fraction turns into a proper fraction, with the numerator having the lower power. But when you factor out the bottom, it switches. I don't know what to do in this situation.