text: Q:
  
  Sqrt induces a problem in simplifying
  
  We have an expression such as 
   Sqrt[-2 - d^2 + 2 Sqrt[1 + d^2]] Sqrt[-d^2 - 2 (1 + Sqrt[1 + d^2])]
  
  When we use 
  FullSimplify[Sqrt[-2 - d^2 + 2 Sqrt[1 + d^2]] Sqrt[-d^2 - 2 (1 + Sqrt[1 + d^2])] ]
  The result is nothing simplified.
  But when we use 
  FullSimplify[(-2 - d^2 + 2 Sqrt[1 + d^2]) (-d^2 - 2 - 2 Sqrt[1 + d^2])]
  
  in which we firstly expanded the second expression and after that we multiplied them to each other albeit without the Sqrt.
  we can see the result is d^4 and Sqrt[d^4] is d^2. 
  How can we directly obtain the the most simplified yield from the first expression i.e.,
  Sqrt[-2 - d^2 + 2 Sqrt[1 + d^2]] Sqrt[-d^2 - 2 (1 + Sqrt[1 + d^2])]
  
  A:
  
  Your purported solution isn't correct. For example:
  Sqrt[-2-d^2+2 Sqrt[1+d^2]] Sqrt[-d^2-2 (1+Sqrt[1+d^2])] /. d->.1
  
  -0.01 + 0. I
  
  It helps to add assumptions:
  FullSimplify[
      Sqrt[-2-d^2+2 Sqrt[1+d^2]] Sqrt[-d^2-2 (1+Sqrt[1+d^2])],
      d ∈ Reals
  ]
  
  -d^2
  
  A:
  
  In:
  expr = Sqrt[-2 - d^2 + 2 Sqrt[1 + d^2]] Sqrt[-d^2 - 2 (1 + Sqrt[1 + d^2])];
  Assuming[d ∈ Reals, Simplify[expr]]
  
  Out:
  
  -d^2
  
  