$\begin{array}{l}\sqrt{x-2\sqrt{x-1}}.(\sqrt{x-1} +1)\\ =\sqrt{x-1 -2\sqrt{x-1} + 1}.(\sqrt{x-1} +1)\\ =\sqrt{(\sqrt{x-1}-1)^2}.(\sqrt{x-1} +1)\\ =|\sqrt{x-1} -1|.(\sqrt{x-1} +1)\\ =\left[\begin{array}{l}(\sqrt{x-1} -1).(\sqrt{x-1} +1)\,\,\,\,Với \,\,x\geq 2\\-(\sqrt{x-1}-1).(\sqrt{x-1} +1)\,\,\,\,Với\,\,x<2\end{array}\right.\\ =\left[\begin{array}{l}x-2\,\,\,\,Với\,\,x\geq2\\2-x\,\,\,\,Với\,\,x<2\end{array}\right.\end{array}$