If [ a + 1/a ]^2 - 2 [ a - 1/a ] = 12, then which of the following is a value of 'a'?

Answer: Option D

$$\begin{array}{rl}& {\left[a+\frac{1}{a}\right]}^2-2\left[a-\frac{1}{a}\right]=12\\ & {\left[a-\frac{1}{a}\right]}^2+4-2\left[a-\frac{1}{a}\right]=12\\ & {\left[a-\frac{1}{a}\right]}^2-2\left[a-\frac{1}{a}\right]-8=0\\ & \text{Let }a-\frac{1}{a}=x\\ & \therefore x^2-2x-8=0\\ & x^2-4x+2x-8=0\\ & \left(x-4\right)\left(x+2\right)=0\\ & x=4,x=-2\\ & a-\frac{1}{a}=-2,a-\frac{1}{a}=4\\ & \therefore a+\frac{1}{a}=\sqrt{4^2+4}\\ & a+\frac{1}{a}=2\sqrt{5}\\ & a-\frac{1}{a}=4\\ & \therefore 2a=2\sqrt{5}+4\\ & a=2+\sqrt{5}\end{array}$$