$a)\ Đk:x\ge0;x\ne1$
$P=\bigg(\dfrac{\sqrt{x}+1}{\sqrt{x}-1}-\dfrac{\sqrt{x}-1}{\sqrt{x}+1}+4\sqrt{x}\bigg).\bigg(\sqrt{x}-\dfrac{1}{\sqrt{x}}\bigg)\\ =\dfrac{x+2\sqrt{x}+1-x+2\sqrt{x}-1+4x\sqrt{x}-4\sqrt{x}}{x-1}.\dfrac{x-1}{\sqrt{x}}\\ =\dfrac{4x\sqrt{x}}{x-1}.\dfrac{x-1}{\sqrt{x}}=4x$
Vậy $P=4x$
$b)x=\dfrac{\sqrt{6}}{2+\sqrt{6}}\Rightarrow P=\dfrac{4\sqrt{6}}{2+\sqrt{6}}\\ =\dfrac{4\sqrt{6}(2-\sqrt{6})}{(2+\sqrt{6})(2-\sqrt{6})}\\ =\dfrac{8\sqrt{6}-24}{4-6}=\dfrac{2(4\sqrt{6}-12)}{-2}=12-4\sqrt{6}$
$c)Để\ \sqrt{P}>P\ thì\ 0<P<1\\ \Leftrightarrow 0<4x<1\\ \Leftrightarrow 0<x<\dfrac{1}{4}$
