<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] Tip:
Notice $1+2x+2x^2+x^3=\left(1+x+x^2\right)(1+x)$
Then $$\begin{split}f(x)&=\frac1{(1-x)^3(1+x)\left(1+x+x^2\right)}\\&=\frac1{1-x}\cdot\frac1{(1-x)(1+x)}\cdot\frac1{(1-x)\left(1+x+x^2\right)}\\&=\frac1{1-x}\cdot\frac1{1-x^2}\cdot\frac1{1-x^3}\\&=\left(1+x+x^2+x^3+...\right)\left(1+x^2+x^4+x^6+...\right)\left(1+x^3+x^6+x^9+...\right)\end{split}$$
Then expansion finishes it all. Try it!
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