Show that the method is of order 3 if a =-5 and of order 2 if a is not equal to -5

Show that the method is of order 3 if a =-5 and of order 2 if a is not
equal to -5

I am studying ODEs and came across this exam question:

I have the solution here also:

I have been working on this exam question all day and have been stuck for
hours. What I don't understand is how the expansion has come about and why
(i.e. hxT and also (xn+h)). Please could someone explain this to me as
this would really really help!
Thank You