\[\begin{array}{l}
Q = \left( {\frac{1}{{\sqrt a + 1}} - \frac{1}{{a + \sqrt a }}} \right):\frac{{\sqrt a - 1}}{{a + 2\sqrt a + 1}}\\
DK:\,\,\,a > 0;\,\,a \ne 1.\\
Q = \left( {\frac{1}{{\sqrt a + 1}} - \frac{1}{{\sqrt a \left( {\sqrt a + 1} \right)}}} \right):\frac{{\sqrt a - 1}}{{{{\left( {\sqrt a + 1} \right)}^2}}}\\
= \frac{{\sqrt a - 1}}{{\sqrt a \left( {\sqrt a + 1} \right)}}.\frac{{{{\left( {\sqrt a + 1} \right)}^2}}}{{\sqrt a - 1}} = \frac{{\left( {\sqrt a - 1} \right)}}{{\sqrt a \left( {\sqrt a + 1} \right)}}.\frac{{{{\left( {\sqrt a + 1} \right)}^2}}}{{\sqrt a - 1}}\\
= \frac{{\sqrt a + 1}}{{\sqrt a }} = 1 + \frac{1}{{\sqrt a }}.\\
\Rightarrow Q \in Z \Leftrightarrow \frac{1}{{\sqrt a }} \in Z \Leftrightarrow \sqrt a \in U\left( 1 \right) \Leftrightarrow a = 1\,\,\,\left( {ktm} \right).\\
d)\,\,Q < - \frac{1}{3}\\
Ta\,\,co:\,\,Q = 1 + \frac{1}{{\sqrt a }} > 0\,\,\,\forall a\,\,tm\,\,DKXD\\
\Rightarrow k\,\,co\,\,a\,\,\,thoa\,\,\,man\,\,\,Q < - \frac{1}{3}.\,\,
\end{array}\]
