To simplify the expression \(\frac{2\sqrt{x} - 3\sqrt{y}}{\sqrt{x} + \sqrt{y}}\), we can multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{x} - \sqrt{y}\).
So we multiply as follows:
\[ \frac{(2\sqrt{x} - 3\sqrt{y})(\sqrt{x} - \sqrt{y})}{(\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})} \]
The denominator can be simplified using the difference of squares:
\[ (\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y}) = x - y \]
Now, we simplify the numerator:
\[ (2\sqrt{x} - 3\sqrt{y})(\sqrt{x} - \sqrt{y}) = 2\sqrt{x} \cdot \sqrt{x} - 2\sqrt{x} \cdot \sqrt{y} - 3\sqrt{y} \cdot \sqrt{x} + 3\sqrt{y} \cdot \sqrt{y} \]
This expands to:
\[ 2x - 2\sqrt{xy} - 3\sqrt{xy} + 3y = 2x - 5\sqrt{xy} + 3y \]
So the whole expression simplifies to:
\[ \frac{2x - 5\sqrt{xy} + 3y}{x - y} \]
Thus, the final simplified form of the expression is:
\[ \frac{2x + 3y - 5\sqrt{xy}}{x - y} \]