jmc

We don't know $g(x)$, so we don't have an expression we can simply stick $25$ in to get an answer. We do, however, know that $g(f(x))=x^2+x+1$. So, if we can figure out what to put into $f(x)$ such that $25$ is the resulting output, we can use our expression for $g(f(x))$ to find $g(25)$.

If $f(x)=25$, then we have $3x^2-2=25$, so $x^2=9$, which means $x=3$ or $x=-3$. Since $x$ could be $3$ or $-3$, we could have $g(25)=g(f(3))$ or $g(25)=g(f(-3))$. Using the given expression for $g(f(x))$, the two possible values of $g(25)$ are $g(f(3))=3^2+3+1=13$ and $g(f(-3))=(-3{)}^2+(-3)+1=7$. The sum of these is $13+7=\boxed{20}$.