Let's simplify the given expression:

$\frac{(1 + \sqrt{5})^2}{2} - \frac{1 - \sqrt{5}}{2}$

Step 1: Expand $(1 + \sqrt{5})^2$

$(1 + \sqrt{5})^2 = 1^2 + 2(1)(\sqrt{5}) + (\sqrt{5})^2 = 1 + 2\sqrt{5} + 5 = 6 + 2\sqrt{5}$

Step 2: Substitute this result back into the expression

$\frac{6 + 2\sqrt{5}}{2} - \frac{1 - \sqrt{5}}{2}$

Step 3: Simplify the fractions

$= \frac{6}{2} + \frac{2\sqrt{5}}{2} - \frac{1}{2} + \frac{\sqrt{5}}{2}$

$= 3 + \sqrt{5} - \frac{1}{2} + \frac{\sqrt{5}}{2}$

Step 4: Combine the terms

$= (3 - \frac{1}{2}) + \left(\sqrt{5} + \frac{\sqrt{5}}{2}\right)$

$= \frac{6}{2} - \frac{1}{2} + \frac{2\sqrt{5}}{2} + \frac{\sqrt{5}}{2}$

$= \frac{5}{2} + \frac{3\sqrt{5}}{2}$

Thus, the simplified form of the expression is:

$\boxed{\frac{5 + 3\sqrt{5}}{2}}$

Would you like more details or further explanations?

Related Questions:

1. How do you expand binomials like $(a + b)^2$?
2. How can you simplify square roots when combining like terms?
3. What are some common mistakes when simplifying radical expressions?
4. How does the distributive property help in expanding expressions?
5. Can this type of expression be simplified further for specific values of $\sqrt{5}$?

Tip: Always combine like terms carefully, especially when dealing with radicals.