Ignore the question for now because you need to understand what the range of validity is for a basic
(1+x)−1.
Expanding this binomially, this is
(1+x)−1=1−x+x2−x3+x4−x5+….
This is the same as writing
(1+x)−1=k=0∑∞(−x)k.
Hopefully you recognise this as an infinite sum of a geometric sequence, where the common ratio between each term is precisely just
−x.
Also hopefully you know that an infinite geometric series gives you a finite value ONLY when the common ratio satisfies
∣r∣<1. This implies that in order for our expansion to be valid and not shoot off to infinity, we require that
∣−x∣<1. This is the same as saying that
∣x∣<1, or in other words,
−1<x<1.
Now you apply this to your question.
In
1+5x6 this is rewritten as
6(1+5x)−1. What is the range of validity here? Compare
(1+5x)−1 with
(1+x)−1. The difference is that we replace
x by
5x in our well understood case of
(1+x)−1 by
5x to obtain
(1+5x)−1.
We do the same and replace
x in
∣x∣<1 to obtain
∣5x∣<1, hence the range of validity is...??
Then you repeat this and determine what the range of validity is for
4(1−3x)−1.
Once you get this point, we can finish up.