1. **State the problem:** Simplify the expression $\left(x - \frac{2}{3}\right) \ln x \ln x$. 2. **Rewrite the expression:** Notice that $\ln x \ln x = (\ln x)^2$. So the expression becomes $$\left(x - \frac{2}{3}\right)(\ln x)^2.$$ 3. **Explain the components:** Here, $x$ is a variable and $\ln x$ is the natural logarithm of $x$. The expression is a product of a linear term in $x$ and the square of the logarithm of $x$. 4. **Simplify if possible:** The expression is already simplified as a product. You can write it as $$x(\ln x)^2 - \frac{2}{3}(\ln x)^2.$$ 5. **Final answer:** $$\boxed{x(\ln x)^2 - \frac{2}{3}(\ln x)^2}.$$