<s>[INST] Theory of Partial Fraction Decomposition The function in question that I want to decompose is $$\dfrac{8x^3 + 7}{(x+1)(2x+1)^3}$$  
I had the idea to to break this down into:
$$\dfrac{A}{x+1} + \dfrac{Bx^2 +Cx + D}{(2x+1)^3} + \dfrac{Ex + F}{(2x+1)^2} + \dfrac{G}{2x+1}$$   
Well this turns into a really messy system of 4 equations and 7 variables.
Is there an easier way to decompose the function? Can I possibly eliminate any of these variables at the outset?
 [/INST] You only need to break it down into $\displaystyle\frac{A}{x+1}$ and $\displaystyle\frac{Bx^2 + Cx + D}{(2x+1)^3}.$ What you end up with is
$\begin{align*}
8A + B &= 8\\
12A + B +C &= 0\\
6A + C + D &= 0\\
A + D &= 7
\end{align*}$
Solving gives $A = 1, B = 0, C = -12, D =6.$ 
Note that this is equivalent to lab bhattacharjee's decomposition.
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