If $\alpha, \beta$ and $\gamma$ are linear operators, establish the identity

$$[\alpha \beta, \gamma]=\alpha[\beta, \gamma]+[\alpha, \gamma] \beta$$

In what follows, the operators $x$ and $p$ are Hermitian and represent position and momentum of a quantum mechanical particle in one-dimension. Show that

$$\left[x^{n}, p\right]=i \hbar n x^{n-1}$$

and

$$\left[x, p^{m}\right]=i \hbar m p^{m-1}$$

where $m, n \in \mathbb{Z}^{+}$. Assuming $\left[x^{n}, p^{m}\right] \neq 0$, show that the operators $x^{n}$ and $p^{m}$ are Hermitian but their product is not. Determine whether $x^{n} p^{m}+p^{m} x^{n}$ is Hermitian.