To simplify the expression \(\frac{\sqrt{5} + 3}{\sqrt{2} + 2}\), we can rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{2} - 2\).
Let's perform the rationalization:
\[ \frac{\sqrt{5} + 3}{\sqrt{2} + 2} \cdot \frac{\sqrt{2} - 2}{\sqrt{2} - 2} \]
This results in:
\[ \frac{(\sqrt{5} + 3)(\sqrt{2} - 2)}{(\sqrt{2} + 2)(\sqrt{2} - 2)} \]
Now, let's simplify the denominator:
\[ (\sqrt{2} + 2)(\sqrt{2} - 2) = (\sqrt{2})^2 - (2)^2 = 2 - 4 = -2 \]
Next, we need to simplify the numerator:
\[ (\sqrt{5} + 3)(\sqrt{2} - 2) = \sqrt{5}\sqrt{2} - 2\sqrt{5} + 3\sqrt{2} - 6 \]
This can be rewritten as:
\[ \sqrt{10} - 2\sqrt{5} + 3\sqrt{2} - 6 \]
Combining the numerator with the denominator gives us:
\[ \frac{\sqrt{10} - 2\sqrt{5} + 3\sqrt{2} - 6}{-2} \]
We can split this into separate terms:
\[ -\frac{\sqrt{10}}{2} + \frac{2\sqrt{5}}{2} - \frac{3\sqrt{2}}{2} + 3 \]
This simplifies to:
\[ -\frac{\sqrt{10}}{2} + \sqrt{5} - \frac{3\sqrt{2}}{2} + 3 \]
Thus, the simplified form of the expression \(\frac{\sqrt{5} + 3}{\sqrt{2} + 2}\) is:
\[ 3 + \sqrt{5} - \frac{3\sqrt{2}}{2} - \frac{\sqrt{10}}{2} \]