a)\(A=\frac{\sqrt{x}+1}{\sqrt{x}-2}+\frac{2\sqrt{x}}{\sqrt{x}+2}+\frac{2+5\sqrt{x}}{4-x}\)

\(=\frac{\sqrt{x}+1}{\sqrt{x}-2}+\frac{2\sqrt{x}}{\sqrt{x}+2}=\frac{2+5\sqrt{x}}{x-4}\)

\(=\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)+2\sqrt{x}\left(\sqrt{x}-2\right)-\left(2+5\sqrt{x}\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(=\frac{x+3\sqrt{x}+2+2x-4\sqrt{x}-2-5\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(=\frac{3x-6\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(=\frac{3\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(=\frac{3\sqrt{x}}{\sqrt{x}+2}\)

b)\(A=2\Leftrightarrow\frac{3\sqrt{x}}{\sqrt{x}+2}=2\)

\(\Leftrightarrow2\left(\sqrt{x}+2\right)=3\sqrt{x}\)

\(\Leftrightarrow2\sqrt{x}+4=3\sqrt{x}\)

\(\Leftrightarrow\sqrt{x}=4\)

\(\Leftrightarrow x=16\)

\(\Rightarrow A\le2\)

Ta có:

\(\frac{x}{y}=\frac{x+1}{y+5}\)

\(\Leftrightarrow y=5x\)

Thế vô P ta được

\(P=\frac{x}{\sqrt{x.5x}+5x}+\frac{5}{\sqrt{x.5x}-x}-\frac{x+5x}{\sqrt{x.5x}}\)

\(=\frac{x}{\sqrt{5}x+5x}+\frac{5x}{\sqrt{5}x-x}-\frac{6x}{\sqrt{5}x}\)

\(=\frac{1}{5+\sqrt{5}}+\frac{5}{\sqrt{5}-1}-\frac{6}{\sqrt{5}}\)

Vậy ta có ĐPCM