Let's simplify the equation step by step:

4^x+1 can be rewritten as 44^x because x+1 = x4^1
So the equation becomes:
44^x - 24^x-2 = 124

Now, we can factor out 4^x from both terms:
4^x(4 - 2*4^-2) = 124

Next, simplify the term inside the parentheses:
4^x(4 - 2/16) = 124
4^x(4 - 1/8) = 124
4^x(32/8 - 1/8) = 124
4^x(31/8) = 124

Finally, divide both sides by 31/8:
4^x = 124 / (31/8)
4^x = 124 * (8/31)
4^x = 32

To solve for x, we need to find the exponent that 4 needs to be raised to in order to equal 32:
x = log4(32)
x = 5/2

Therefore, the value of x is 5/2.