  We improve an $L^2\times L^2\to L^2$ estimate for a certain bilinear operator
in the finite field of size $p$, where $p$ is a prime sufficiently large. Our
method carefully picks the variables to apply the Cauchy-Schwarz inequality. As
a corollary, we show that there exists a quadratic progression $x,x+y,x+y^2$
for nonzero $y$ inside any subset of $\mathbb{F}_p$ of density $\gtrsim
p^{-1/8}$
