for part a, how do we come to conclusions on which method to use

Now for this one, there are lots of techniques available. I notice that the two denominators happen to be conjugates of each other, so combining into a single fraction is possible.

Moreover, the two numerators are the same, meaning that there is going to be some simplification ensuing after that.

And the best part is, after combining into a single fraction, the denominator can be expanded into two terms, and just nice the two-term combination is an identity which can be simplified into a single term. This allows the numerator and the denominator to be simplified even more easily.

For these reasons, this is a good approach to look at.