On simplification, $$\frac{x^3 - y^3}{x[(x + y)^2 - 3xy]} \div \frac{y[(x - y)^2 + 3xy]}{x^3 + y^3} \times \frac{(x + y)^2 - (x - y)^2}{x^2 - y^2}$$ is equal to:

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

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

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

= $$\frac{x^2 + xy + y^2}{x[x^2 - xy + y^2]} \times \frac{x^2 - xy + y^2}{y[x^2 + xy + y^2]} \times 4xy$$ = 4