Which expression below gives the average rate of change of the function h(x)= 4^x+2 on the interval -2 x 2

I believe your interval is [-2,2]:  all numbers between -2 and +2.

Then the ave. r. of c. on this interval is:

f(2) - f(-2)      4^2 + 2 - {4^(-2) + 2}
=
  2 - (-2)                      4

                       16 - (1/16)            15 15/16
                 = = = 3.98 or 255/64
                                4                        4