jmc

We begin by factoring both quadratics. We find that the $18x^2+25x-3=0$ factors to: $$(2x+3)(9x-1)=0.$$Thus, the only values of $x$ that satisfy this equation are $-\frac{3}{2}$ and $\frac{1}{9}$. When we factor the second quadratic, $4x^2+8x+3=0$, we find that it is equal to: $$(2x+1)(2x+3)=0.$$Thus, the only values that satisfy this equation are $-\frac{1}{2}$ and $-\frac{3}{2}$. Since $-\frac{3}{2}$ is the only root both polynomials have in common, the answer must be $\boxed{-\frac{3}{2}}$