check your arithmetic! you dropped a factor of 3 from the term -3[8-k-6] (you should have -3[2-k] at the end, not -[2-k]).
as far as finding P goes, i believe that dim(ker(A-I)) = 1 (prove this!), so the Jordan form of A is:
now A-I has nilpotency 3, right? so we can pick any vector in R^3 for an element in ker((A-I)^3).
it appears your text has chosen v1 = e1, and then chosen
v2= (A-I)v1, which is in ker((A-I)^2), but not in ker((A-I)^3), so...