Decompose[poly, x] decomposes a polynomial, if possible, into a composition of simpler polynomials.
Decompose gives a list of the polynomials which can be composed as to give the original polynomial.
The set of polynomials is not necessarily unique.
Decomposition is an operation which is independent of polynomial factorization.
See The Mathematica Book: Section 3.3.4.
See also: FactorList, Solve.