\(P_{\left(x\right)}=\dfrac{2\sqrt{x}}{x+2\sqrt{x}}+\dfrac{x-1}{x+3\sqrt{x}+2}\) \(\left(x>0\right)\)
\(=\dfrac{2\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+2\right)}+\dfrac{x-1}{x+\sqrt{x}+2\sqrt{x}+2}\)
\(=\dfrac{2}{\sqrt{x}+2}+\dfrac{x-1}{\sqrt{x}\left(\sqrt{x}+1\right)+2\left(\sqrt{x}+1\right)}\)
\(=\dfrac{2}{\sqrt{x}+2}+\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{2}{\sqrt{x}+2}+\dfrac{\sqrt{x}-1}{\sqrt{x}+2}\)
\(=\dfrac{\sqrt{x}+1}{\sqrt{x}+2}\)
Vậy \(P_{\left(x\right)}=\dfrac{\sqrt{x}+1}{\sqrt{x}+2}\) với \(x>0\)
