the only constant polynomial satisfying the given is P(x)=0 for all x.
let P(x)= $\sum a_{i}x^i$ where i runs through 0 ,1,...,n
then putting it into the main equation given , we get that
$a_{n}x^{3n}+........+a_1.x^3+a_0$+$a_n.x^{2n}+......+a_0$
= $a_n.(x^3+1)^n+.....+a_1(x^3+1)+a_0$...(*)
as (*) is an identity , then we equate the constant terms and after cancelling out equal terms from LHS & RHS ,we get that ,
$a_n.x^{2n}+.....+a_1.x^2$=$a_{n}.n.x^{3n-3}$+ a polynomial expression of x whose degree is < [3(n-1)]
now if n>3 , then the degree of R.H.S > that of L.H.S. , so we must have n is less than or equal to 3.
so , the rest is easy