To find the product $R(x) = f(x) \cdot g(x)$ of the rational functions $f(x) = \frac{3x - 21}{x^2 - 9x + 14}$ and $g(x) = \frac{2x^2 - 8}{3x + 6}$, apply the standard procedure for multiplying rational expressions:

Step 1. Factor the numerator and denominator of each rational expression completely: $R(x) = \frac{3(x - 7)}{(x - 7)(x - 2)} \cdot \frac{2(x - 2)(x + 2)}{3(x + 2)}$

Step 2. Multiply the resulting fractions together: $R(x) = \frac{3 \cdot 2(x - 7)(x - 2)(x + 2)}{3(x - 7)(x - 2)(x + 2)}$

Step 3. Simplify the expression by dividing out the common factors of $3$, $(x - 7)$, $(x - 2)$, and $(x + 2)$: $R(x) = 2$

The simplified product of the rational functions is $2$.