To perform multiple operations on rational expressions, such as $\frac{3x-6}{4x-4} \cdot \frac{x^2+2x-3}{x^2-3x-10} \div \frac{2x+12}{8x+16}$, first rewrite the division as multiplication by the reciprocal: $\frac{3x-6}{4x-4} \cdot \frac{x^2+2x-3}{x^2-3x-10} \cdot \frac{8x+16}{2x+12}$. Next, factor all numerators and denominators to obtain $\frac{3(x-2)}{4(x-1)} \cdot \frac{(x+3)(x-1)}{(x-5)(x+2)} \cdot \frac{8(x+2)}{2(x+6)}$. Then, multiply the fractions together, bringing constants to the front to help identify numerical factors: $\frac{3 \cdot 8(x-2)(x+3)(x-1)(x+2)}{4 \cdot 2(x-1)(x-5)(x+2)(x+6)}$. Finally, simplify by dividing out the common factors of $8$, $(x-1)$, and $(x+2)$ from both the numerator and denominator, resulting in the simplified expression $\frac{3(x-2)(x+3)}{(x-5)(x+6)}$.