Expanding One Function in Powers of Another One sees here that it is possible to expand $f(x) = 2x^3 + 7x^2 + x - 6$ in powers of $x - 2$ by taylor expanding $f(x) = f(x - 2 + 2) = f(2 + h)$ about $2$, and this idea can be used in deriving the form of a partial fraction expansion whose denominator contains powers of linear factors, e.g. here.
But what about expanding $f$ in powers of $(ax^2+bx+c)$ to derive the form of a partial fractions expansion containing powers of a quadratic in the denominator, e.g. here?
Is the only method to do this Burmann's theorem?
 A: When $g(x)$ is not a linear polynomial it might not be true that one can expand $f(x)$ as a linear combination of powers of $g(x)$. For example: $f(x)=x$ is not a linear combination of powers of $g(x):=x^2$.
If the problem has a solution, i.e. $f$ is a linear combination of powers of $g(x)$, then you can proceed as follows:
Assume we have a solution for the problem
$$f(x):=a_n[g(x)]^n+a_{n-1}[g(x)]^{n-1}+...+a_1g(x)+a_0$$
If $x=a$ is a point where $g(x)$ vanishes, then 
$$\begin{align}f(a)&=a_0\\\frac{f(x)-a_0}{g(x)}|_{x=a}&=a_1\\\frac{f(x)-a_0-a_1g(x)}{g(x)}|_{x=a}&=a_2\\\frac{f(x)-a_0-a_1g(x)-a_2[g(x)^2]}{g(x)}|_{x=a}&=a_3\\\vdots &\phantom{{}={}}\vdots\\\frac{f(x)-a_0-...-a_k[g(x)]^k}{g(x)}|_{x=a}&=a_{k+1}\end{align}$$
Notice that the terms can be found successively beginning from the first equation.
In the case of $f$ and $g$ being polynomials, one can use Ruffinit-Horner algorithm to evaluate the left-hand sides. Ruffini-Horner is a method for evaluating polynomials at a number, but as a by-product it gives you the value of the polynomial in the left-hand side of the next equation.
If you need this for partial fraction decomposition, then you don't really need to consider expanding in powers of quadratic polynomials. You can obtain the partial fraction decomposition over the complex numbers, by completely factoring all the denominator into linear factors, which is always possible over the complex. After obtaining the partialfraction decomposition over the complex, you can add the summads that are complex conjugates to each other and this gives you the partial fraction decomposition over the reals.
