If a + 1/a + 1 = 0 ( a 0 ) , then the value of ( a^4 - a )\, is?

Answer: Option A

$$\begin{array}{rl}& a+\frac{1}{a}+1=0\\ & a+\frac{1}{a}=-1\\ & \text{Squaring both sides}\\ & \Rightarrow a^2+\frac{1}{a^2}+2=1\\ & \Rightarrow a^2+\frac{1}{a^2}=-1\\ & \Rightarrow a^2+1=-\frac{1}{a^2}.....(i)\\ & \Rightarrow a+\frac{1}{a}=-1\left(\text{Given}\right)\\ & \therefore a^2+1=-a.....(ii)\\ & \Rightarrow -a=\frac{-1}{a^2}\\ & \text{For equation (i) and (ii)}\\ & {\text{a}}^3=1\\ & \therefore {\text{a}}^3-1=0\\ & \Rightarrow a^4-a=a\left(a^3-1\right)\\ & \Rightarrow a^4-a=a\times 0\\ & \Rightarrow a^4-a=0\end{array}$$