Polynomial $f(x)$ satisfies $(x - 1)f(x + 1) - (x + 2)f(x) = 0$ for every $x \in \mathbb{R}$. If $f(2) = 6$, find the value of $f\left(\frac{3}{2}\right)$. The original answer is in \frac{k}{m} format, please give the value of k + m.

Let's think step by step and output the final answer within \boxed{}.
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