Polynomial expansion concerns the heuristic expectation that, for a typical polynomial \( P \) in \( n \) variables over a field \( F \) and subsets \( A_1,\ldots,A_n \) of \( F \), the image \( P(A_1,\ldots,A_n) \) is substantially larger than each of the individual sets \( A_k \). We establish new expansion results for certain classes of polynomials over finite fields, including a classification result for ternary quadratic polynomials. Our methods rely on spectral bounds for certain graphs arising from incidence geometry. This is joint work with Sam Chow.