jmc

When we factor $3x^2+nx+72$, our two factors are of the form $(3x+A)(x+B)$, where $A$ and $B$ are integers. We must have $AB=72$, and we want $3B+A$ to be as large as possible (because $3B+A$ is the coefficient of $x$ when $(3x+A)(x+B)$ is expanded). We make $3B+A$ as large as possible by letting $B=72$ and $A=1$; any other possibility reduces $3B$ much more than $A$ increases. Therefore, the largest possible value of $n$ is $3B+A=3(72)+1=\boxed{217}$.