Systems, Matrices, and Determinants · Challenge problem

The partial fraction decomposition $\dfrac{2x^2-5x+1}{x(x-1)(x+1)}=\dfrac{A}{x}+\dfrac{B}{x-1}+\dfrac{C}{x+1}$ is really a $3\times3$ linear system for $A$, $B$, $C$ in disguise. Find $A+B+C$.