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 $\displaystyle (p+qx^2)^r$ up to and including the term in x^4 where $\displaystyle p, q, r \in \mathbb{R}$

The key gives the following answer:

$\displaystyle p^r(1+\frac{p}{q}x^2)=p^r(1+r\frac{q}{p}x^2+\frac{ r(r-1)}{2}\frac{q^2}{p^2}x^4)$

I understand how they can move out $\displaystyle p^r$ 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 $\displaystyle x^n$ 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?