Sorry about the title. I couldn't think of anything else.
I am currently looking through a worked solution to a problem and came across a factorization I don't understand. The two steps are below. I don't see how (2) was obtained from (1).
$$(1) \ \ \ 5 \cdot 3^k - 5 \cdot 2^k - 6 \cdot 3^{k-1} + 6 \cdot 2^{k-1}$$ $$(2) \ \ \ 3^{k-1}(5 \cdot 3 - 6) - 2^{k-1}(5 \cdot 2 - 6) \ \ \ \ $$
Sometimes factorizations aren't obvious, and the only way to see them is to work them backwards, and practice. For this one, let's just check that they're in fact equal. $(2)$ below gives:
$$3^{k-1}(5 \cdot 3 - 6) - 2^{k-1}(5 \cdot 2 - 6) = 5 \cdot 3^k - 6 \cdot 3^{k-1} - 5 \cdot 2^k - 6 \cdot 2^{k-1}$$
This is $(1)$ with the terms written in a different order.
In order to guess this factorization yourself, one might try to remove a maximal number of powers of $3$ possible from those terms with $3$s explicitly (not $6$s here). In this case, that number happens to be the $k-1$ power, so we pull that many out. Then what is left is none in the $6$, but one $3$ remains attached to the $5$. The story is the same for the $2$s.