A Clean Separation

Let $f(x) = \ln x + x^2 - kx + 1$ with $k \in \mathbb{R}$, and $g(x) = x^2 - 3x + x\mathrm{e}^x$.

1. Find the intervals on which $f(x)$ is monotonic, discussing all cases. 2. If the inequality $f(x) \leqslant g(x)$ holds for all $x$ in the domain, find the range of possible values of $k$.