Solution
= Solution
For the <partial fraction decomposition>, multiply by $(x-1)(x-2)$ to obtain the <polynomial identity>
$$
x^2+1=a(x-1)(x-2)+b(x-2)+c(x-1).
$$
Comparing the $x^2$ coefficients gives $a=1$. Substitution of $x=1$ gives $2=-b$, while $x=2$ gives $5=c$. Therefore
$$
\boxed{a=1,\qquad b=-2,\qquad c=5.}
$$
The original identity holds for $x\ne1,2$; evaluating the multiplied <polynomial identity> at those values is valid.