Divide $X$ into two sets $X_1$ and $X_2$ of equal size ($n/2$), and recursively construct polynomials $p_1(n)$ and $p_2(n)$, such that $p_1(X_1)=0$ and $p_2(X_2)=0$. Then use the FFT algorithm to compute the polynomial $p_1\cdot p_2$. This algorithm satisfies the recurrence $T(n)=2T(n/2)+bn\log n$, for some constant $b$, which implies that $T(n)$ is $O(n\log n)$ by the master theorem.