\(a,\text{ ĐKXĐ : }x>1\)
\(P=\left(\frac{-8\sqrt{x}-8}{x+2\sqrt{x}-3}-\frac{\sqrt{x}+3}{1-\sqrt{x}}\right):\left(2-\frac{\sqrt{x}+4}{\sqrt{x}+3}\right)\)
\(=\left[\frac{-8\sqrt{x}-8}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}+\frac{\left(\sqrt{x}+3\right)^2}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\right]:\frac{\sqrt{x}+2}{\sqrt{x}+3}\)
\(=\frac{-8\sqrt{x}-8+x+6\sqrt{x}+9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}.\frac{\sqrt{x}+3}{\sqrt{x}+2}\)
\(=\frac{\left(x-2\sqrt{x}+1\right)\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)
\(=\frac{\left(\sqrt{x}-1\right)^2\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)\(=\frac{\sqrt{x}-1}{\sqrt{x}+2}\)
\(\text{b,}\)
\(P< \frac{1}{2}\Leftrightarrow\frac{\sqrt{x}-1}{\sqrt{x}+2}< \frac{1}{2}\)
\(\Leftrightarrow\frac{\sqrt{x}-1}{\sqrt{x}+2}-\frac{1}{2}< 0\)
\(\Leftrightarrow\frac{2\sqrt{x}-2-\sqrt{x}-2}{2\left(\sqrt{x}+2\right)}< 0\)
\(\Leftrightarrow\frac{\sqrt{x}-4}{2\left(\sqrt{x}+2\right)}< 0\)
\(\Leftrightarrow\sqrt{x}-4< 0\)
\(\Leftrightarrow1< x< 16\)
\(c,\)
\(\frac{1}{P}\in Z\Leftrightarrow\frac{\sqrt{x}-1}{\sqrt{x}+2}\in Z\)
\(\Leftrightarrow\frac{\sqrt{x}+2-3}{\sqrt{x}+2}\in Z\)
\(\Leftrightarrow1-\frac{3}{\sqrt{x}+2}\in Z\)
\(\Leftrightarrow\sqrt{x}+2\inƯ_3=\left\{1;3\right\}\)
\(\Rightarrow x=1\)
