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?

$$f(x)(1+2x+2x^2+x^3) = \frac{1}{(1-x)^3}\implies $$ $$(1-x-x^2+x^4+x^5-x^6)\,f(x)=1$$
Now, let $f(x)=a+b x+c x^2$ and ignore the terms $x^n$ if $n>3$. This gives
$$a+ (b-a)x- (a+b-c)x^2=1$$
Comparing the terms, $a=1$, $b=a=1$, $c=a+b=2$.
