Answer
$-1, \frac{1}{2}$.
Work Step by Step
$\frac{x-3}{x^2+1}=-2 \Longrightarrow x-3=-2x^2-2 \Longrightarrow 2x^2+x-1=0 \Longrightarrow (2x-1)(x+1)=0 \Longrightarrow x=-1, \frac{1}{2}$. Check, both fit.
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