Does the polynomial belong to the ideal
(Assuming you are working in $\mathbb{F}[x]$, with $\mathbb{F}$ a field)
By definition of the $\gcd$, the ideal generated by $1-x^3$ and $1+2x+2x^2+x^3$ is the ideal generated by the $\gcd$ of these two polynomials. Here, you get that $$\langle 1-x^3, 1+2x+2x^2+x^3\rangle=\langle x^2+x+1\rangle$$
The polynomial $x^3$ is clearly not a multiple of $x^2+x+1$, so it does not belong to the ideal generated by $1-x^3$ and $1+2x+2x^2+x^3$.