To evaluate the product function $(f \cdot g)(2)$ for $f(x) = x + 2$ and $g(x) = x^2 - 3x - 4$, first determine the general product function, which is $(f \cdot g)(x) = x^3 - x^2 - 10x - 8$. Next, substitute $x = 2$ into this resulting polynomial: $(f \cdot g)(2) = 2^3 - 2^2 - 10(2) - 8$. Simplify the expression according to the order of operations: $8 - 4 - 20 - 8$, which sums to $-24$. Thus, $(f \cdot g)(2) = -24$.