it's there

take the polynomial out to x^{8}or so and look at the first 7 terms, ignore the 8th

you should see 1 + x + 3/2x^{2}+ 1/2x^{3}+ 1/8x^{4}+ ....

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.