\(\begin{array}{l}
(\frac{1}{{\sqrt a + 1}} - \frac{1}{{a\sqrt a }})(\frac{{\sqrt a - 1}}{{a + 2\sqrt a + 1}})\\
= \frac{{a\sqrt a - \sqrt a - 1}}{{a\sqrt a (\sqrt a + 1)}}.\frac{{\sqrt a - 1}}{{{{(\sqrt a + 1)}^2}}}\\
= \frac{{{a^2} - a\sqrt a - a + \sqrt a - \sqrt a + 1}}{{a\sqrt a {{(\sqrt a + 1)}^3}}}\\
= \frac{{{a^2} - a\sqrt a - a + 1}}{{a\sqrt a {{(\sqrt a + 1)}^3}}}\\
= \frac{{a\sqrt a (\sqrt a - 1) - (\sqrt a - 1)(\sqrt a + 1)}}{{a\sqrt a {{(\sqrt a + 1)}^3}}}\\
= \frac{{(a\sqrt a - \sqrt a - 1)(\sqrt a - 1)}}{{a\sqrt a {{(\sqrt a + 1)}^3}}}
\end{array}\)
