Rút gọn biểu thức \(P = \left( {1 + \frac{{\sqrt x }}{{x + \sqrt x + 1}}} \right):\frac{{\sqrt x + 1}}{{x\sqrt x - 1}}\) (x ≥ 0, x ≠ 1).

Với x ≥ 0, ta có:

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

\(P = \frac{{x + \sqrt x + 1 + \sqrt x }}{{x + \sqrt x + 1}}.\frac{{x\sqrt x - 1}}{{\sqrt x + 1}}\)

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

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

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

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