take the polynomial out to x8 or so and look at the first 7 terms, ignore the 8th
you should see 1 + x + 3/2x2 + 1/2x3 + 1/8x4 + ....
you should see this as 1 + x + Sum[2,Infinity, some pattern]
The sum will correspond to a well known power series minus a couple terms. You're also adding a couple terms at the front.
Collect all that together and you'll end up with an expression for the solution as a function of x.