To solve the expression \(4 \cdot (3 + 1) \cdot (3^2 + 1) \cdot (3^4 + 1) \cdot (3^8 + 1) \cdot (3^{16} + 1)\), we can simplify it step by step.

First, let's simplify each term inside the parentheses:

1. \(3 + 1 = 4\)
2. \(3^2 + 1 = 9 + 1 = 10\)
3. \(3^4 + 1 = 81 + 1 = 82\)
4. \(3^8 + 1 = 6561 + 1 = 6562\)
5. \(3^{16} + 1 = 43046721 + 1 = 43046722\)

Now, substitute these values back into the expression:

\[4 \cdot 4 \cdot 10 \cdot 82 \cdot 6562 \cdot 43046722\]

Next, multiply these values together:

1. \(4 \cdot 4 = 16\)
2. \(16 \cdot 10 = 160\)
3. \(160 \cdot 82 = 13120\)
4. \(13120 \cdot 6562 = 86008320\)
5. \(86008320 \cdot 43046722 = 3696930995863040\)

Thus, the value of the expression is \(3696930995863040\).