1. **State the problem:** Simplify the expression $$\frac{3 + \sqrt{5} + \sqrt{44}}{2}$$. 2. **Recall the rules:** The square root of a product can be simplified as $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$. Also, addition inside the numerator can be combined after simplification. 3. **Simplify the square root:** $$\sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11}$$. 4. **Rewrite the expression:** $$\frac{3 + \sqrt{5} + 2\sqrt{11}}{2}$$. 5. **Final answer:** The expression is simplified as $$\frac{3 + \sqrt{5} + 2\sqrt{11}}{2}$$. This is the simplest exact form.