A shortcut to the Maclaurin series expansion?

(Not sure if this should go into the calculus section; feel free to move it.)

**Question:** Find the Maclaurin series for up to and including the term in x^4 where

The key gives the following answer:

I understand how they can move out and expand the parentheses binomially to get the right hand side of the equation, but there are two things that puzzle me:

1. The question asks for me to find a Maclaurin series for the expression. To me, that means finding the first derivatives and of the expression, dividing them by a factorial and multiplying by an term. This is not done here. Why does the answer given correspond to a Maclaurin series without performing these steps?

2. In which cases is it valid to do the above trick? When should one think of using it?