\begin{align*} &(1-x)y''+xy'-y=-(x-1)^2&\Rightarrow\\ &(1-x)y''+y'+(x-1)y'-y=-(x-1)^2&\Rightarrow\\ &\frac{(1-x)y''+y'}{(1-x)^2}-\frac{(1-x)y'+y}{(1-x)^2}=-1&\Rightarrow\\ &\left(\frac{y'}{1-x}\right)'-\left(\frac{y}{1-x}\right)'=-1&\Rightarrow\\ &\frac{y'}{1-x}-\frac{y}{1-x}=-x+C_1&\Rightarrow\\ &y'-y=x^2-(C_1+1)x+C_1&\Rightarrow\\ &e^{-x}y'-e^{-x}y=x^2e^{-x}-(C_1+1)xe^{-x}+C_1e^{-x}&\Rightarrow\\ &(e^{-x}y)'=x^2e^{-x}-(C_1+1)xe^{-x}+C_1e^{-x}&\Rightarrow\\ &e^{-x}y=\int \left[x^2e^{-x}-(C_1+1)xe^{-x}+C_1e^{-x}\right] dx+C_2&\Rightarrow\\ &e^{-x}y=-e^{-x}(x^2-C_1x+x+1)+C_2&\Rightarrow\\ &y=-x^2+C_1x-x-1+C_2e^x. \end{align*}
Am I Right? Do you have better solutions?