You did not distribute when multiplying the numerator by the denominator's conjugate. Multiply the numerator and denominator by the conjugate of the denominator:

$\displaystyle\frac{{{x}}}{{{\left({x}+\sqrt{{{3}}}\right)}}}\cdot\frac{{{\left({x}-\sqrt{{{3}}}\right)}}}{{{\left({x}-\sqrt{{{3}}}\right)}}}\usepackage{cmbright}\Huge\$

Use the Distributive Property and FOIL and simplify radicals.