Answer
$\{-1,-\frac{1}{3}\}$.
Work Step by Step
As $9=3^2,\ 27=3^3$, we have $3^{4x}\cdot 3^{3x^2}=3^{-1}$. Thus we have $3x^2+4x=-1$ or $3x^2+4x+1=0$, which gives $x=-1,-\frac{1}{3}$ or $\{-1,-\frac{1}{3}\}$. Checking answers, they are both good.
