To rationalize the given expression \(\frac{2\sqrt{3} + \sqrt{5}}{\sqrt{5} - \sqrt{3}}\), we can multiply both the numerator and the denominator by the conjugate of the denominator (\(\sqrt{5} + \sqrt{3}\)):

\(

\frac{2\sqrt{3} + \sqrt{5}}{\sqrt{5} - \sqrt{3}} \cdot \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} = \frac{(2\sqrt{3} + \sqrt{5})(\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})}

\)

Simplify the numerators and denominators:

\(

\frac{2\sqrt{3} \cdot \sqrt{5} + 2\sqrt{3} \cdot \sqrt{3} + \sqrt{5} \cdot \sqrt{5} + \sqrt{5} \cdot \sqrt{3}}{5 - 3}

\)

\(

\frac{2\sqrt{15} + 6 + 5 + \sqrt{15}}{2}

\)

Combine like terms:

\(

\frac{2\sqrt{15} + \sqrt{15} + 6 + 5}{2}

\)

\(

\frac{3\sqrt{15} + 11}{2}

\)

So, after rationalizing the expression, we get \(\frac{3\sqrt{15} + 11}{2}\).