\(Q=\left(\dfrac{2\sqrt{x}}{\sqrt{x}\left(x+1\right)-\left(x+1\right)}-\dfrac{1}{\sqrt{x}-1}\right):\cdot\dfrac{x+\sqrt{x}+1}{x+1}\\ =\left(\dfrac{2\sqrt{x}-x-1}{\left(\sqrt{x-1}\right)\left(x+1\right)}\right).\dfrac{x+1}{x+\sqrt{x}+1}\\ =\dfrac{-\left(x-2\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)}.\dfrac{1}{x+\sqrt{x}+1}=\dfrac{-\left(\sqrt{x}-1\right)^2}{\sqrt{x}-1}.\dfrac{1}{x+\sqrt{x}+1}=\dfrac{-\sqrt{x}+1}{x+\sqrt{x}+1}\)
\(Q=\left(\dfrac{2\sqrt{x}}{x\sqrt{x}+\sqrt{x}-x-1}-\dfrac{1}{\sqrt{x}-1}\right):\left(1+\dfrac{\sqrt{x}}{x+1}\right)\left(x\ge0;x\ne1\right)\)
\(=\left[\dfrac{2\sqrt{x}}{\sqrt{x}\left(x+1\right)-\left(x+1\right)}-\dfrac{1}{\sqrt{x}-1}\right]:\dfrac{x+\sqrt{x}+1}{x+1}\)
\(=\left[\dfrac{2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+1\right)}-\dfrac{x+1}{\left(\sqrt{x}-1\right)\left(x+1\right)}\right].\dfrac{x+1}{x+\sqrt{x}+1}\)
\(=\dfrac{2\sqrt{x}-x-1}{\left(\sqrt{x}-1\right)\left(x+1\right)}.\dfrac{x+1}{x+\sqrt{x}+1}\)
\(=\dfrac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(x+1\right)}.\dfrac{x+1}{x+\sqrt{x}+1}\)
\(=\dfrac{\sqrt{x}-1}{x+\sqrt{x}+1}\)
