Homework: Maclaurin Power Series Help I'm trying to find the Maclaurin Power Series for
$$f(x)=\frac{3x-8}{3x^2+5x-2}$$
but each degree of differentiation gets more complex with no discernible pattern. Any help is appreciated, thanks.

$$\begin{align*} f(x)&=\frac{3x-8}{3x^2+5x-2}=\frac{3x-8}{(x+2)(3x-1)}=\{\text{partial fraction decomposition}\}=\\
&=\frac{2}{2+x}-\frac{3}{3x-1}=\frac{1}{1-\left(-\frac{x}{2}\right)}+\frac{3}{1-3x}=\{\text{geometric series for }|x|<\frac{1}{3}\}=\\
&=\sum_{n=0}^\infty\left(-\frac{x}{2}\right)^n+3\sum_{n=0}^\infty(3x)^n=\sum_{n=0}^\infty\left(\left(-\frac{x}{2}\right)^n+3(3x)^n\right)=\\
&=\sum_{n=0}^\infty\left(\frac{1}{(-2)^n}+3^{n+1}\right)x^n \end{align*}$$
