Square root of x+8= square root of x-4 then add 2

Square root of x+8= square root of x-4 then add 2 has several ambiguities, not the least in that you haven't provided the instructions.  I'm making the assumption that you want to solve    Square root of x+8= square root of x-4 then add 2    for x:

√(x+8) = √(x-4)   +   2

Square both sides:

x+8 = (x-4) + 4√(x-4) + 4, or
  4    =   4√(x-4) + 4, or 0= 4√(x-4), or 4√(x-4)
 = 0,
or
√(x-4) = 0.  Squaring both sides, we get x-4=0, so that x=4.  It is essential to check solutions; substitute 4 for x in the original problem and determine whether the equation is true or false.