Ta có: \(\left\lbrack\left(x+1\right)\sqrt{y}+\sqrt{x}\right\rbrack\left(\sqrt{x}-\sqrt{y}\right)+\left\lbrack\left(y+1\right)\sqrt{x}-\sqrt{y}\right\rbrack\left(\sqrt{x}+\sqrt{y}\right)\)
\(=\sqrt{xy}\left(x+1\right)-y\left(x+1\right)+x-\sqrt{xy}+\left(y+1\right)\cdot x+\left(y+1\right)\cdot\sqrt{xy}-\sqrt{xy}-y\)
\(=\sqrt{xy}\left(x+y+2\right)-xy-y+xy+x+x-y-2\sqrt{xy}\)
\(=\sqrt{xy}\left(x+y\right)+2x-2y\)
\(=\sqrt{\left(5+\sqrt{21}\right)\left(5-\sqrt{21}\right)}\left(5+\sqrt{21}+5-\sqrt{21}\right)+2\left(5+\sqrt{21}-5+\sqrt{21}\right)\)
\(=\sqrt4\cdot10+2\cdot2\sqrt{21}=20+4\sqrt{21}\)
Ta có: \(P=\frac{\left(x+1\right)\cdot\sqrt{y}+\sqrt{x}}{\sqrt{x}+\sqrt{y}}+\frac{\left(y+1\right)\cdot\sqrt{x}-\sqrt{y}}{\sqrt{x}-\sqrt{y}}\)
\(=\frac{\left\lbrack\left(x+1\right)\sqrt{y}+\sqrt{x}\right\rbrack\left(\sqrt{x}-\sqrt{y}\right)+\left\lbrack\left(y+1\right)\sqrt{x}-\sqrt{y}\right\rbrack\left(\sqrt{x}+\sqrt{y}\right)}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}\)
\(=\frac{20+4\sqrt{21}}{x-y}=\frac{20+4\sqrt{21}}{2\sqrt{21}}=\frac{10+2\sqrt{21}}{\sqrt{21}}=\frac{10\sqrt{21}+42}{21}\)
