Substituting $x=1-x$, we have $f(1-x)+(1-x)f(x)=x^2-2x+2$ Multiplying this by $x$, we have $xf(1-x)+x(1-x)f(x)=x^3-2x^2+2x$ Subtracting our original equation from this, we have $f(x)+xf(1-x)-[xf(1-x)+x(1-x)f(x)]=x^2+1-(x^3-2x^2+2x)$ $f(x)-x(1-x)f(x)=-x^3+3x^2-2x+1$ $[1-x(1-x)]f(x)=-x^3+3x^2-2x+1$ $f(x)=\frac{-x^3+3x^2-2x+1}{x^2-x+1}$