If a function \( f : (-\infty, 2) \to \mathbb{R} \) is defined by \[ f(x)= \begin{cases} \dfrac{\alpha \left|x^2 - 3x + 2\right|}{x - 1}, & x < 1 \\[8pt] \dfrac{\sin([x] - x)}{x - [x]}, & x > 1 \\[8pt] \beta, & x = 1 \end{cases} \] and \( f \) is continuous at \( x = 1 \), then find \[ \frac{\alpha^2 + \beta^2}{|\alpha \beta|}. \]
The area of the region \( \{(x, y): 0 \leq y \leq x^2 + 1, \, 0 \leq y \leq x + 1, \, 0 \leq x \leq 2\ \) is:}