Answer
$\dfrac{a-4\sqrt{a}+3}{a-1}$
Work Step by Step
Multiplying by the conjugate of the denominator and using $(a+b)(a-b)=a^2-b^2,$ the given expression, $
\dfrac{\sqrt{a}-3}{\sqrt{a}+1}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt{a}-3}{\sqrt{a}+1}\cdot\dfrac{\sqrt{a}-1}{\sqrt{a}-1}
\\\\=
\dfrac{\sqrt{a}(\sqrt{a})+\sqrt{a}(-1)-3(\sqrt{a})-3(-1)}{(\sqrt{a})^2-(1)^2}
\\\\=
\dfrac{a-1\sqrt{a}-3\sqrt{a}+3}{a-1}
\\\\=
\dfrac{a-4\sqrt{a}+3}{a-1}
.\end{array}
