jmc

We can factor $6x^2+x-2$ as $(3x+2)(2x-1)$. For this quantity to be negative, one of the factors must be positive and one of the factors must be negative.

The first factor, $3x+2$, is zero at $x=-\frac{2}{3}$. It's negative for $x<-\frac{2}{3}$ and positive for $x>-\frac{2}{3}$.

The second factor, $2x-1$, is zero at $x=\frac{1}{2}$. It's negative for $x<\frac{1}{2}$ and positive for $x>\frac{1}{2}$.

The interval on which one factor is positive and the other is negative is $-\frac{2}{3}<x<\frac{1}{2}$. In this interval, the largest integer (indeed, the only integer!) is $x=\boxed{0}$.