\(P=1-\left(\frac{\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}+\frac{\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\left(\frac{2}{\sqrt{x}}-\frac{2-x}{\sqrt{x}\left(\sqrt{x}+1\right)}\right)\)
\(=1-\left(\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\frac{\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\left(\frac{2\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}-\frac{2-x}{\sqrt{x}\left(\sqrt{x}+1\right)}\right)\)
\(=1-\left(\frac{x+2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\left(\frac{x+2\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+1\right)}\right)\)
\(=1-\frac{\left(x+2\sqrt{x}\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}.\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(x+2\sqrt{x}\right)}\)
\(=1-\frac{\sqrt{x}}{\sqrt{x}-1}=-\frac{1}{\sqrt{x}-1}\)
