You can rearrange to $\displaystyle 2 \ln x - 3x^2 - 6x \le a$, then find the max value for the left hand side in the interval x>0. That max value is the value for a.
You can rearrange to $\displaystyle 2 \ln x - 3x^2 - 6x \le a$, then find the max value for the left hand side in the interval x>0. That max value is the value for a.