Let \( \mathbb{F}_q \) be a finite field of order \( q \), and let \( P \) be a polynomial in \( \mathbb{F}_q[x_1, x_2] \). For a set \( A \subset \mathbb{F}_q \), define \( P(A) := \{ P(x_1, x_2) \mid x_i \in A \} \). Using certain constructions of expanders, we characterize all polynomials \( P \) for which the following holds:

If \( |A + A| \) is small (compared to \( |A| \)), then \( |P(A)| \) is large.

The case \( P = x_1 x_2 \) corresponds to the well-known sum-product problem.