Consider random polynomials of the form $\sum_{i=0}^n a_i x^i$, where the coefficients $a_i$ are independent with mean zero and finite second moments. We discuss how the probability that such a polynomial has no zeros decays as $n^{-b+o(1)}$, where $b$ is a universal constant independent of the distributions of $a_i$. This resolves a conjecture by Poonen and Stoll. Notable progress on this conjecture was made by Dembo, Poonen, Shao, and Zeitouni, who demonstrated the same result when coefficients are identically and independently distributed (i.i.d.) with all higher moments being finite. Extending their results to the finite moment case remained an open problem until now. Our approach, in contrast to previous works in this area, relies on the combinatorics of multiscale analysis of random polynomials. This work is based on joint collaboration with Sumit Mukherjee.