Consider the quadratic polynomial W = 3x^2+3x+1. Then, it is easy
(!) to prove by induction on n, the following polynomial
(3x+1)^(3n+2) +3x+2 == 3(3nx+3n+1)W ( mod 3W^2 ).
Putting x = n gives the desired divisibility property (with the
additional factor 3).
I prefer to ommit the details of the induction proof. I simply
mention that I made use of the following helpfull identities :
(3x+1)^3 = 9xW+1 and x(3x+2) = w-x-1.