jmc

The vertical asymptotes will occur when the denominator of a simplified rational expression is equal to zero. We factor the denominator $3x^2+5x+2$ to obtain $(3x+2)(x+1)$. Hence, there are vertical asymptotes when $x=-1,-\frac{2}{3}$, and the sum of these values of $x$ is $-1-\frac{2}{3}=\boxed{-\frac{5}{3}.}$

(We can also use Vieta's formulas, which states that the sum of the roots of $a{x}^2+bx+c=0$ is $-b/a$.)