If $$x = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}  and  y = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$$, the value of $$\frac{x^2 + xy + y^2}{x^2 - xy + y^2}$$ is

 $$x = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}  and  y = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$$

$$ x+y = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} + \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$$

$$x+y =\frac{5+1+2\sqrt{5} + 5 + 1 -2\sqrt{5}}{5-1} $$

$$x+y = \frac{12}{4}$$ = 3

$$x\times y= \frac{\sqrt{5} + 1}{\sqrt{5} - 1} \times \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$$ = 1

$$(x+y)^2 = x^2 +y^2 +2xy$$= $$(3)^2 = x^2 +y^2 +2$$

$$ x^2 +y^2 = 7 $$

 $$\frac{x^2 + xy + y^2}{x^2 - xy + y^2}$$ {substituting $$x^2 +y^2 = 7 and xy =1$$

 $$\frac{7+1}{7-1}$$ = $$\frac{8}{6}$$ 

= $$\frac{4}{3}$$