a.
\(P=\left(\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}+1}-\dfrac{x+2}{\sqrt{x}+1}\right):\left(\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\dfrac{\sqrt{x}-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)
\(=\left(\dfrac{x+\sqrt{x}-x-2}{\sqrt{x}+1}\right):\left(\dfrac{x-\sqrt{x}+\sqrt{x}-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)
\(=\left(\dfrac{\sqrt{x}-2}{\sqrt{x}+1}\right):\left(\dfrac{x-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)
\(=\left(\dfrac{\sqrt{x}-2}{\sqrt{x}+1}\right):\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{\left(\sqrt{x}-2\right)}{\left(\sqrt{x}+1\right)}.\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{\sqrt{x}-1}{\sqrt{x}+2}\)
b.
\(P=\dfrac{\sqrt{x}+2-3}{\sqrt{x}+2}=1-\dfrac{3}{\sqrt{x}+2}\ge1-\dfrac{3}{2}=-\dfrac{1}{2}\)
\(P_{min}=-\dfrac{1}{2}\) khi \(\sqrt{x}=0\Rightarrow x=0\)
c.
\(Q=\left(\dfrac{\sqrt{x}-1}{\sqrt{x}+2}\right).\left(2\sqrt{x}+x\right)=\dfrac{\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+2\right)}.\sqrt{x}\left(\sqrt{x}+2\right)\)
\(=\sqrt{x}\left(\sqrt{x}-1\right)=x-\sqrt{x}\)
\(=\left(x-\sqrt{x}+\dfrac{1}{4}\right)-\dfrac{1}{4}=\left(\sqrt{x}-\dfrac{1}{2}\right)^2-\dfrac{1}{4}\ge-\dfrac{1}{4}\)
\(Q_{min}=-\dfrac{1}{4}\) khi \(\sqrt{x}-\dfrac{1}{2}=0\Rightarrow x=\dfrac{1}{4}\)