To find the product $R(x) = f(x) \cdot g(x)$ of the rational functions $f(x) = \frac{2x - 6}{x^2 - 8x + 15}$ and $g(x) = \frac{x^2 - 25}{2x + 10}$, follow the procedure for multiplying rational expressions:

Step 1. Factor each numerator and denominator completely: $R(x) = \frac{2(x - 3)}{(x - 3)(x - 5)} \cdot \frac{(x - 5)(x + 5)}{2(x + 5)}$

Step 2. Multiply the numerators and denominators together: $R(x) = \frac{2(x - 3)(x - 5)(x + 5)}{2(x - 3)(x - 5)(x + 5)}$

Step 3. Simplify the rational expression by dividing out the common factors of $2$, $(x - 3)$, $(x - 5)$, and $(x + 5)$: $R(x) = 1$

Thus, the product of the two rational functions simplifies to $1$.