Q:

how to solve for the unknowns given two equating polynomials in mathematica

I have, for example, the following polynomials to be equated, and I need to determine the unknowns c1, c2, c3, respectively. How can I do this automatically in mma, especially when there are many terms involved?
x+2*x^3+4*x^5==(c1+c2)*(x+2*c2*x^3)+(4-c1)*c3*x^5

Many thanks.
Edit: ideally, I want to equate the coefficients of left and right for the terms with equal exponents in x. Then solve this system of equations.

A:

If this has to be true for all x, you could use SolveAlways (not tested)
SolveAlways[x+2*x^3+4*x^5==(c1+c2)*(x+2*c2*x^3)+(4-c1)*c3*x^5, x]

