When $(3 + 2x + x^{2})(1 + mx + m^{2}x^{2})$ is expanded, the terms that include an $x^{2}$ will come from multiplying a constant with a term that includes $x^{2}$ or multiplying two terms that includes $x$. In other words, the term that includes $x^{2}$ will be $3 \cdot m^{2} x^{2} + 2x \cdot mx + x^{2} \cdot 1 = (3m^{2} + 2m + 1)x^{2}$. From the condition that the coefficient of this term equals 1, we see that $3m^{2} + 2m + 1 = 1$ which gives $3m^{2} + 2m = 0$ or $m(3m + 2) = 0$, which means that $m = 0$ or $m = -\frac{2}{3}$. The sum of these possible values of $m$ is $-\frac{2}{3}$.