We are asked to simplify the expression $A = (1 + \frac{5 - \sqrt{5}}{1 - \sqrt{5}})(1 + \frac{5 + \sqrt{5}}{1 + \sqrt{5}})$.

1. Problem Description

We are asked to simplify the expression $A = (1 + \frac{5 - \sqrt{5}}{1 - \sqrt{5}})(1 + \frac{5 + \sqrt{5}}{1 + \sqrt{5}})$.

2. Solution Steps

First, let's simplify the term $\frac{5 - \sqrt{5}}{1 - \sqrt{5}}$. We can rewrite $5$ as $\sqrt{5} \cdot \sqrt{5} \cdot \sqrt{5} \cdot \sqrt{5}/5$ or $\sqrt{5}^4 / 5$, so $5 = \sqrt{5} \cdot \sqrt{5}$. Therefore, we have

$$\frac{5 - \sqrt{5}}{1 - \sqrt{5}} = \frac{\sqrt{5}(\sqrt{5} - 1)}{1 - \sqrt{5}} = \frac{-\sqrt{5}(1 - \sqrt{5})}{1 - \sqrt{5}} = -\sqrt{5}.$$

Thus, $1 + \frac{5 - \sqrt{5}}{1 - \sqrt{5}} = 1 - \sqrt{5}$.

Next, let's simplify the term $\frac{5 + \sqrt{5}}{1 + \sqrt{5}}$. Similarly, we have

$$\frac{5 + \sqrt{5}}{1 + \sqrt{5}} = \frac{\sqrt{5}(\sqrt{5} + 1)}{1 + \sqrt{5}} = \frac{\sqrt{5}(1 + \sqrt{5})}{1 + \sqrt{5}} = \sqrt{5}.$$

Thus, $1 + \frac{5 + \sqrt{5}}{1 + \sqrt{5}} = 1 + \sqrt{5}$.

Finally, we can calculate $A$:

$A = (1 - \sqrt{5})(1 + \sqrt{5})$. We can use the difference of squares formula, which is

$$(a - b)(a + b) = a^2 - b^2.$$

So, we have

$$A = (1 - \sqrt{5})(1 + \sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4.$$

3. Final Answer

The final answer is -4.