What is the big O for $n^{\log n} + n!$?
Look at this
$$(\frac{n}{e})^{\log{n}} = \frac{n^{\log{n}}}{n} = n^{{\log{n}}-1}$$
And compare it with $(\frac{n}{e})^n$ (as $e^{\log{n}} = n$, because of $\log$ function property). By multiplying $n$ to the above eqautation you will get $n^{\log{n}}$. Hence, just we need to compare $n$ and $(\frac{n}{e})^{n-\log{n}}$. So, definitely, we will find that the growth of $n!$ is much higher than $n^{\log{n}}$.