Maybe as a passing comment:
I don't like how the marking scheme has written the solution. Aside from the the lack of words, which seems to be something I complain a lot about, the key ingredient in induction is to show where you actually use the induction assumption by literally writing the phrase "by induction assumption" on the correct line (or if you're lazy like me, write "I.A." on the correct equal sign), not at the end.
Also, I tend to start the induction step with "we would like to show that 'blah'", just to know what the goal is.
Sometime (and in fact more often than not) you could get lost in a bunch of calculations that you just forgot what you're trying to achieve. Writing down your goal often gives you a hint as to what your next logical step should be.
For instance in this case, you have in the induction step:
If we know explicitly what your goal is, i.e.
, there is a strong hint that factorizing is a good idea. Of course now it comes down to how good are you at factorizing expressions... (again, borrowing from calculus, people often say calculus is not hard, it's the algebra)