Minimum Value of Expression with Domain Restriction

Let $t = 3 - x$. Since $x < 3$, we have $t > 0$. Express $x$ in terms of $t$: $x = 3 - t$.

Substitute into the expression: $\frac{x^2 - 6x + 10}{3 - x} = \frac{(x - 3)^2 + 1}{3 - x} = \frac{(-t)^2 + 1}{t} = \frac{t^2 + 1}{t} = t + \frac{1}{t}$

For $t > 0$, by AM-GM inequality, $t + \frac{1}{t} \ge 2$. The minimum possible value is $2$, which occurs when $t = 1$, i.e., $x = 2$.