\( a)\,\left( {1 - \sqrt x } \right)\left( {1 + \sqrt x + x} \right) \)
\( = \left( {1 - \sqrt x } \right)\left[ {1 + 1\sqrt x + {{\left( {\sqrt x } \right)}^2}} \right] \)
\( = 1 - {\left( {\sqrt x } \right)^3} = 1 - x\sqrt x \) (với \(x \ge 0\))
\( b)\,\left( {\sqrt x + 2} \right)\left( {x - 2\sqrt x + 4} \right) \)
\( = \left( {\sqrt x + 2} \right)\left[ {{{\left( {\sqrt x } \right)}^2} - \sqrt x .2 + {2^2}} \right] \)
\( = {\left( {\sqrt x } \right)^3} + {2^3} = x\sqrt x + 8\) (với \(x \ge 0\))
c) \(\left( {\sqrt x - \sqrt y } \right)\left( {x + y + \sqrt {xy} } \right)\)
\( = \left( {\sqrt x - \sqrt y } \right)\left[ {{{\left( {\sqrt x } \right)}^2} + \sqrt x .\sqrt y + {{\left( {\sqrt y } \right)}^2}} \right]\)
\( = {\left( {\sqrt x } \right)^3} - {\left( {\sqrt y } \right)^3} = x\sqrt x - y\sqrt y \) (với \(x \ge 0\), \(y \ge 0\))
\( d)\,\,\left( {\sqrt x + \sqrt y } \right)\left( {{x^2} + y - x\sqrt y } \right) \)
\( = \left( {\sqrt x + \sqrt y } \right)\left[ {{x^2} - x\sqrt y + {{\left( {\sqrt y } \right)}^2}} \right] \)
\( = {x^3} + {\left( {\sqrt y } \right)^3} = {x^3} + y\sqrt y \) (với \(y \ge 0\))
