If $$x = 2 + \sqrt5$$ then the value of $$x^3 + x^{-3}$$ is:

Given, $$x=2+\sqrt{5}$$

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

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

$$=$$> $$\ \frac{1}{x}=\ \frac{2-\sqrt{5}}{4-5}$$

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

$$\therefore\ x+\frac{1}{x}=(2+\sqrt{5})+(\sqrt{5}-2)=2\sqrt{5}$$

$$=$$> $$\left(\ x+\frac{1}{x}\right)^3=(2\sqrt{5})^3$$

$$=$$> $$x^3+\frac{\ 1}{x^3}+3.x.\frac{1}{x}\left(\ x+\frac{1}{x}\right)=8\left(5\sqrt{5}\right)$$

$$=$$> $$x^3+\frac{\ 1}{x^3}+3\left(2\sqrt{5}\right)=40\sqrt{5}$$

$$=$$> $$x^3+x^{-3}=34\sqrt{5}$$

Hence, the correct answer is Option A