<s>[INST] Find variables from a multiplication of geometric sequences 
Given $$f(x) = \sum_{i=0}^\infty a_ix^i$$
And $$f(x)(1+2x+2x^2+x^3) = \frac{1}{(1-x)^3}$$
Find the values of $a_0$, $a_1$, and $a_2$.

I try to expand them but it doesn't seem to lead me to any solution.
How can I isolate a0, a1, and a2 to find them?
 [/INST] $f(x)= a_0+a_1x+a_2x^2+O(x^3)$
$f(x)(1+2x+2x^2+x^3)$
$=a_0+(2a_0+a_1)x+(2a_0+2a_1+a_2) x^2
+O(x^3)$
$$\textrm{Again } f(x)(1+2x+2x^2+x^3)=\frac{1}{(1-x)^3}=1+3x+6x^2+O(x^3)$$
Comparing the like powers of x we get:$$a_0=1,a_1=1,a_2=2$$
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