Follow this 3 easy steps, it's not that hard as it seems..
#x^3 - 3b^(2/3) x + 9a#
Where #x = (2a + sqrt(4a^2 - b^2))^(1/3) + (2a - sqrt(4a^2 - b^2))^(1/3)#
Step 1 #-># Substitute the value of x into the main equation..
#color(red)x^3 - 3b^(2/3) color(red)(x) + 9a#
#color(red) [[(2a + sqrt(4a^2 - b^2))^(1/3) + (2a - sqrt(4a^2 - b^2))^(1/3)]]^3 - 3b^(2/3) color(red)[[(2a + sqrt(4a^2 - b^2))^(1/3) + (2a - sqrt(4a^2 - b^2))^(1/3)] + 9a#
Step 2 #-># Eliminating the powers..
#[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))]^(1/3 xx cancel3) - 3b^(2/3) [(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))]^(1/3) + 9a#
#[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))] - 3b^(2/3) [(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))]^(1/3) + 9a#
#[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))] - 3b[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))]^((1/3) xx (2/3)) + 9a#
#[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))] - 3b[(2a + sqrt(4a^2 - b^2)) + (2a - sqrt(4a^2 - b^2))]^(2/9) + 9a#
Step 3 #-># Collecting like terms..
#(2a + 2a) + (sqrt(4a^2 - b^2) - sqrt(4a^2 - b^2)) - 3b[(2a + 2a) + (sqrt(4a^2 - b^2) - sqrt(4a^2 - b^2))]^(2/9) + 9a#
#4a + cancel(sqrt(4a^2 - b^2) - sqrt(4a^2 - b^2)) - 3b[4a + cancel(sqrt(4a^2 - b^2) - sqrt(4a^2 - b^2))]^(2/9) + 9a#
#rArr# #4a + 0 - 3b[4a + 0]^(2/9) + 9a#
#rArr# #4a - 3b[4a]^(2/9) + 9a#
#rArr# #4a + 9a - (3b xx 4a)^(2/9)#
#rArr# #13a - 12ab^(2/9) -> Answer#
Hence Option 4 is the suitable Answer..
