Positive on a Closed Interval

Let $f(x) = x^2 - ax\ln x + a + 1$ with $a \in \mathbb{R}$.

1. When $a = 1$, find the equation of the tangent line to $f(x)$ at the point $(1, f(1))$. 2. If $f(x) > 0$ for every $x \in [1, \mathrm{e}]$, find the range of possible values of $a$.