If $$P = \frac{x^4 - 8x}{x^3 - x^2 - 2x}, Q = \frac{x^2 + 2x + 1}{x^2 - 4x - 5}$$ and $$R = \frac{2x^2 + 4x + 8}{x - 5}$$, then $$(P \times Q) \div R$$ is equal to:

$$P = \frac{x^4 - 8x}{x^3 - x^2 - 2x}$$

$$Q = \frac{x^2 + 2x + 1}{x^2 - 4x - 5}$$

= $$\frac{(x + 1)^2}{x^2 - 4x - 5 + 9 - 9}$$

$$(P \times Q) \div R$$

= $$(\frac{x^4 - 8x}{x^3 - x^2 - 2x} \times \frac{x^2 + 2x + 1}{x^2 - 4x - 5}) \div \frac{2x^2 + 4x + 8}{x - 5}$$

= $$\frac{x^4 - 8x}{x^3 - x^2 - 2x} \times \frac{x^2 + 2x + 1}{x^2 - 4x - 5} \times \frac{x - 5}{2x^2 + 4x + 8}$$

= $$\frac{x(x^3 - 8)}{x^3 - x^2 - 2x} \times \frac{x^2 + 2x + 1}{x^2 - 4x - 5} \times \frac{x - 5}{2(x^2 + 2x + 4)}$$

= $$\frac{x(x - 2)(x^2 + 2x - 4)}{x(x^2 - x - 2)} \times \frac{(x + 1)^2}{x^2 - 5x + x - 5} \times \frac{x - 5}{2(x^2 + 2x + 4)}$$

= $$\frac{(x - 2)}{(x^2 - 2x + x- 2)} \times \frac{(x + 1)^2}{(x +1)(x - 5)} \times \frac{x - 5}{2}$$

= $$\frac{(x - 2)}{(x - 2)(x + 1)} \times \frac{(x + 1)}{2}$$

= $$\frac{1}{2}$$