We multiplied \(A\) by \(-3x\) which shifts every term over one spot and multiplies them by \(-3\text{.}\) On the third line, we multiplied \(A\) by \(2x^2\text{,}\) which shifted every term over two spots and multiplied them by 2. When we add up the corresponding terms, we are taking each term, subtracting 3 times the previous term, and adding 2 times the term before that. This will happen for each term after \(a_1\) because \(a_n - 3a_{n-1} + 2a_{n-2} = 0\text{.}\) In general, we might have two terms from the beginning of the generating series, although in this case the second term happens to be 0 as well.
in-context