Partial Fractions Are Fun!

$\displaystyle \begin{aligned} \frac{x^2 + 2}{(x - 1)(x - 2)(x - 3)} & = \frac{A}{x - 1} + \frac{B}{x - 2} + \frac{C}{x - 3} & \small \color{#3D99F6} \text{Multiply throughout by } (x - 1)(x - 2)(x - 3) \\ \implies A(x - 2)(x - 3) + B(x - 1)(x - 3) + C(x - 1)(x - 2) & = x^2 + 2 & \small \color{#3D99F6} \text{Let } x = 2 \\ \implies B(2 - 1)(2 - 3) & = 2^2 + 2 \\ \implies B & = \boxed{- 6} \end{aligned}$

After finding $A$ and $C$ in a similar fashion, our decomposed fraction is

$\displaystyle \begin{aligned} \frac{3}{2(x - 1)} - \frac{6}{x - 2} + \frac{11}{2(x - 3)} \end{aligned}$