If x = (√2 + 1)/(√2 - 1), then what is the value of $$(x^5 + x^4 + x^2 + x)/x^3$$ ?

Given : $$x=\frac{\sqrt2+1}{\sqrt2-1}$$

=> $$x=\frac{\sqrt2+1}{\sqrt2-1}\times\frac{(\sqrt2+1)}{(\sqrt2+1)}$$

=> $$x=\frac{(\sqrt2+1)^2}{2-1}$$

=> $$x=3+2\sqrt2$$ -------------(i)

Squaring both sides, we get :

=> $$x^2=(3+2\sqrt2)^2$$

=> $$x^2=17+12\sqrt2$$ ------------(ii)

To find : $$\frac{x^5+x^4+x^2+x}{x^3}$$

= $$\frac{x^4+x^3+x+1}{x^2}$$

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

Substituting values from equations (i) and (ii)

= $$\frac{(17+12\sqrt2)(17+12\sqrt2+3+2\sqrt2)+(3+2\sqrt2+1)}{17+12\sqrt2}$$

= $$\frac{(17+12\sqrt2)(20+14\sqrt2)+(4+2\sqrt2)}{17+12\sqrt2}$$

= $$\frac{(340+238\sqrt2+240\sqrt2+336)+(4+2\sqrt2)}{17+12\sqrt2}$$

= $$\frac{680+480\sqrt2}{17+12\sqrt2}$$

= $$\frac{40(17+12\sqrt2)}{17+12\sqrt2}=40$$

=> Ans - (A)