First, add #color(red)(8)# to each side of the equation to isolate the radical term while keeping the equation balanced:

#2sqrt(x - 11) - 8 + color(red)(8) = 4 + color(red)(8)#

#2sqrt(x - 11) - 0 = 12#

#2sqrt(x - 11) = 12#

Next, divide each side of the equation by #color(red)(2)# to isolate the radical while keeping the equation balanced:

#(2sqrt(x - 11))/color(red)(2) = 12/color(red)(2)#

#(color(red)(cancel(color(black)(2)))sqrt(x - 11))/cancel(color(red)(2)) = 6#

#sqrt(x - 11) = 6#

Then square each side of the equation to eliminate the radical while keeping the equation balanced:

#(sqrt(x - 11))^2 = 6^2#

#x - 11 = 36#

Now, add #color(red)(11)# to each side of the equation to solve for #x# while keeping the equation balanced:

#x - 11 + color(red)(11) = 36 + color(red)(11)#

#x - 0 = 47#

#x = 47#

To check the solution substitute #color(red)(47)# for #color(red)(x)# in the original equation and calculate the left side of the equation to ensure it equals #4#

#2sqrt(color(red)(x) - 11) - 8 = 4# becomes:

#2sqrt(color(red)(47) - 11) - 8 = 4#

#2sqrt(36) - 8 = 4#

#(2 xx +-6) - 8 = 4#

#+-12 - 8 = 4#

#4 = 4#

Or

#-20 != 4#

The solution of #-20# is extraneous.