Let $a$ and $b$ be positive integers,
$$\begin{aligned}
& (x - a)(x + 2a) < 0 \\
& (x - b)(x + 2b) > 0
\end{aligned}$$
Given this system of inequalities.
If $a + b = 8$ and the solution set of this system of inequalities contains 16 integers, what is the product $a \cdot b$?
A) 7
B) 10
C) 12
D) 15
E) 16