I would have tried multiplying the numerator and the denominator by
the conjugate of the denominator.
But do this instead for less work. Factor out a square root of three in
the numerator and see what you can cancel.
\(\displaystyle \dfrac{3 \ + \ \sqrt{3}}{1 \ + \ \sqrt{3}} \ =\)
\(\displaystyle \dfrac{\sqrt{3}( \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ )}{ \ \ \ \ \ 1 \ + \ \sqrt{3}} \ = \ ?\)