Let $a$ and $b$ be real numbers. The functions f and g are defined on the set of real numbers as
$$\begin{aligned}
& f(x) = ax - b \\
& g(x) = bx - 2
\end{aligned}$$
Given that
$$\begin{aligned}
& (f + g)(1) = f(1) \\
& (f + g)(2) = g(2)
\end{aligned}$$
what is the product $\mathbf{a} \cdot \mathbf{b}$?
A) 2
B) 4
C) 6
D) 8
E) 10