If $$P = \frac{x^3 + y^3}{(x - y)^2 + 3xy}, Q = \frac{(x + y)^2 - 3xy}{x^3 - y^3}$$ and $$R = \frac{(x + y)^2 + (x - y)^2}{x^2 - y^2}$$, then what is the value of $$(P \div Q) \times R$$?

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

= $$(\frac{x^3 + y^3}{(x - y)^2 + 3xy} \div \frac{(x + y)^2 - 3xy}{x^3 - y^3}) \times \frac{(x + y)^2 + (x - y)^2}{x^2 - y^2}$$

= $$\frac{x^3 + y^3}{(x - y)^2 + 3xy} \times \frac{x^3 - y^3}{(x + y)^2 - 3xy} \times \frac{(x + y)^2 + (x - y)^2}{x^2 - y^2}$$

= $$\frac{(x + y)(x^2 - xy + y^2)}{x^2 + y^2 - 2xy + 3xy} \times \frac{(x - y)(x^2 + xy + y^2)}{x^2 + y^2 + 2xy - 3xy} \times \frac{x^2 + y^2 + 2xy + x^2 + y^2 - 2xy}{(x + y)(x - y)}$$

= $$\frac{(x^2 - xy + y^2)}{x^2 + xy + y^2} \times \frac{(x^2 + xy + y^2)}{x^2 - xy + y^2} \times 2(x^2 + y^2) $$

= $$ 2(x^2 + y^2) $$