To simplify the expression \( B = (3 - \sqrt{5}) \sqrt{3 + \sqrt{5}} + (3 + \sqrt{5}) \sqrt{3 - \sqrt{5}} \), we can evaluate each term separately and then simplify.

Let's consider the first term:
\[ (3 - \sqrt{5}) \sqrt{3 + \sqrt{5}}. \]

Now perform a similar operation with the second term:
\[ (3 + \sqrt{5}) \sqrt{3 - \sqrt{5}}. \]

To simplify these expressions, we look at the product of the terms under the square roots:
\[ \sqrt{(3 - \sqrt{5})(3 + \sqrt{5})} \]
and
\[ \sqrt{(3 + \sqrt{5})(3 - \sqrt{5})}. \]

Using the difference of squares formula, which states that \( (a - b)(a + b) = a^2 - b^2 \), we get:
\[ (3 - \sqrt{5})(3 + \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4. \]

So, both of the square root expressions simplify to \(\sqrt{4} = 2\).

Substituting these back into the original expression:
\[ B = (3 - \sqrt{5})(2) + (3 + \sqrt{5})(2). \]

Expanding this, we get:
\[ B = 2(3 - \sqrt{5}) + 2(3 + \sqrt{5}). \]

Distribute the 2:
\[ B = (6 - 2\sqrt{5}) + (6 + 2\sqrt{5}). \]

Now combine like terms:
\[ B = 6 - 2\sqrt{5} + 6 + 2\sqrt{5}. \]

The \(-2\sqrt{5}\) and \(+2\sqrt{5}\) cancel each other out, leaving:
\[ B = 6 + 6 = 12. \]

Thus, the value of \( B \) is \( 12 \).