TITLE: Prove that an expression is divisible by a polynomial
QUESTION [5 upvotes]: Question: Show that the polynomial $f(x)=(x+1)^{2n} +(x+2)^n  -  1$
is divisible by $g(x) = x^2+3x+2$, where $n$ is an integer.
I have tried to use mathematical induction. The basis case wasn't that difficult, but when it comes to the inductive step itself, I got a bit confused.
Is it possible to prove this by mathematical induction, and is binominal expansion required at that step?

REPLY [4 votes]: Hint $\rm\: mod\ g\!:\,\ e_i^2\! = e_i\Rightarrow\: e_1^n + e_2^n =\, e_1\! + e_2.\ $ Google (orthogonal) idempotents to learn more.