In a problem, I noticed the author did this:
$$\frac{1}{(a+2)+(z-2)} = \frac{1}{(a+2)}\cdot \frac{1}{1+\frac{z-2}{a+2}}$$
What he is saying is to take the entire $(a+2)$ term and multiply it by $1$ and then also to $\frac{z-2}{a+2}$ in order to get the same thing as on the left. But growing up, I learned that whenever we needed to multiply $(a+b)\cdot (c+d)$ we needed to do $(a\cdot c+ a\cdot d +b\cdot c + b\cdot d)$. Why is it ok to do what the author did instead?
The author is simply factoring the denominator, and the fact that there are sums in the parentheses is not relevant. A simpler version of the same thing: $\dfrac{1}{\phantom{\big(}A+B\phantom{\big)}}=\dfrac{1}{A\left(1+\frac{B}{A}\right)}=\dfrac{1}{\phantom{\big(}A\phantom{\big(}}\cdot\dfrac{1}{\left(1+\frac{B}{A}\right)}$