<s>[INST] Partial fractions decomposition of ${\frac{2x}{(x+2)^2}}$ Express in partial fraction form: $\displaystyle{\frac{2x}{(x+2)^2}}$
I think is $\displaystyle{\frac{2x}{(x+2)^{2}} = \frac{A}{x+2}+\frac{B}{(x+2)^2}}$
However when identifying $A$ and $B$, I'm not sure how to calculate A.
E.g.  $$2x = A\cdot (x+2) + B$$
Substitute $x=-2$    
$2\cdot(-2)$ = $A\cdot (2-2) +B$
$-4 = B$
In other questions there is always another factor to multiply by at this stage.
 [/INST] $$2x = A(x+2) + B$$
You just found that $B=-4$ and now you want to find the value of $A$. Well, let $B=-4$ and see what happens...
\begin{align}
   2x &= A(x+2)-4 \\
   2x+4 &= A(x+2) \\
   2(x+2) &= A(x+2)
\end{align}
Since this has to be true for all $x$, then $A=2$.
This is not a lucky coincidence. Substituting back into the original equation will always result in something solvable.
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