jmc

Rearranging $x^2-x-1=0$ gives $x^2=x+1$. So, repeatedly substituting $x+1$ for $x^2$ gives us $$\begin{array}{rl}{x}^3-2x+1& =x(x^2)-2x+1\\ & =x(x+1)-2x+1\\ & =x^2+x-2x+1\\ & =x^2-x+1\\ & =(x+1)-x+1\\ & =\boxed{2}\end{array}$$