Z=(X+2/√X+1 - √x ):(√x - 4/ 1-x - √x/ √x +1

$\begin{array}{l} Z = \left( {\frac{{x + 2}}{{\sqrt x + 1}} - \sqrt x } \right):\left( {\frac{{\sqrt x - 4}}{{1 - x}} - \frac{{\sqrt x }}{{\sqrt x + 1}}} \right)\,\,\,\left( {DK:\,\,x \ge 0;\,\,x \ne 1} \right)\\ = \frac{{x + 2 - x - \sqrt x }}{{\sqrt x + 1}}:\frac{{\sqrt x - 4 - \sqrt x \left( {1 - \sqrt x } \right)}}{{\left( {1 - \sqrt x } \right)\left( {1 + \sqrt x } \right)}}\\ = \frac{{2 - \sqrt x }}{{\sqrt x + 1}}:\frac{{\sqrt x - 4 - \sqrt x + x}}{{\left( {1 - \sqrt x } \right)\left( {1 + \sqrt x } \right)}}\\ = \frac{{2 - \sqrt x }}{{\sqrt x + 1}}:\frac{{x - 4}}{{\left( {1 - \sqrt x } \right)\left( {1 + \sqrt x } \right)}}\\ = \frac{{2 - \sqrt x }}{{\sqrt x + 1}}.\frac{{\left( {1 - \sqrt x } \right)\left( {1 + \sqrt x } \right)}}{{\left( {\sqrt x - 2} \right)\left( {\sqrt x + 2} \right)}}\\ = \frac{{1 - \sqrt x }}{{ - \left( {\sqrt x + 2} \right)}} = \frac{{\sqrt x - 1}}{{\sqrt x + 2}}. \end{array}$