jmc

The general term in the expansion of ${\left(\frac{3}{2}{x}^2-\frac{1}{3x}\right)}^6$ is $$\left(\genfrac{}{}{0ex}{}{6}{k}\right){\left(\frac{3}{2}{x}^2\right)}^k{\left(-\frac{1}{3x}\right)}^{6-k}=\left(\genfrac{}{}{0ex}{}{6}{k}\right){\left(\frac{3}{2}\right)}^k{\left(-\frac{1}{3}\right)}^{6-k}{x}^{3k-6}.$$To get the constant term, we take $k=2,$ which gives us $$\left(\genfrac{}{}{0ex}{}{6}{2}\right){\left(\frac{3}{2}\right)}^2{\left(-\frac{1}{3}\right)}^4=\boxed{\frac{5}{12}}.$$