Take the four numbers as (a-3d),(a-d),(a+d),(a+3d) in AP with common difference as 2d.
Now given that Sum of four numbers = 96
i.e (a-3d)+(a-d)+(a+d)+(a+3d) = 96
=> 4a = 96
=> a = 24 ----------(1)
Now to find d use the other condition.
i.e Product of means/Product of Extremes = 35/27
=> (a-d)(a+d) / (a-3d)(a+3d) = 35/27
=> a2 - d2 / a2 -9d2 = 35/27 ---------------(2)
Now find the value of d using equation (1) and (2)
You will get d = 4
So the four numbers are:
a-3d = 24-3*4 = 12
a-d = 24 - 4 = 20
a+d = 24+4 = 28
a+3d = 24+12 = 36
Hence the four numbers are 12,20,28,36