Hermitian or Not?

x and p are both Hermitian, since they correspond to observables. Note that $p = -i\hbar \partial_x$ , so the given operator is actually proportional to $-\hat{x}^2\hat{p}^2$ . Taking the Hermitian conjugate and using the Hermitian property of x and p, we find the conjugate to be $-\hat{p}^2 \hat{x}^2$ , which is not the same.

Given operator is actually $\frac{-1}{\hslash} \hat{x}^2\hat{p}^2$ . Since both $\hat{x}^2 \text{ and } \hat{p}^2$ are Hermitian, their product would be if they commuted. It's easy to check that they don't, which is expected since $\hat{x}$ and $\hat{p}$ don't commute.