To find the product $R(x) = f(x) \cdot g(x)$ of the rational functions $f(x) = \frac{x^2 - x}{3x^2 + 27x - 30}$ and $g(x) = \frac{x^2 - 100}{x^2 - 10x}$, use the procedure for multiplying rational expressions:

Step 1. Factor each numerator and denominator completely. Remember to factor out the greatest common factor first, if applicable: $R(x) = \frac{x(x - 1)}{3(x + 10)(x - 1)} \cdot \frac{(x - 10)(x + 10)}{x(x - 10)}$

Step 2. Multiply the numerators and denominators together: $R(x) = \frac{x(x - 1)(x - 10)(x + 10)}{3x(x + 10)(x - 1)(x - 10)}$

Step 3. Simplify the expression by dividing out the common factors of $x$, $(x - 1)$, $(x - 10)$, and $(x + 10)$. Since all factors in the numerator cancel, a $1$ remains in the numerator, while a $3$ remains in the denominator: $R(x) = \frac{1}{3}$