Answer:
Let’s break this apart 
Well we know the median has to be 4
Since it’s 5 numbers the middle number has to be 4 since its the median. 
Let’s put in what we know.
a, b, 4, d, e
Constraints:
There has to be more then 1 “4”. 
e-a = 4
So using that information lets solve since the possibility is almost endless 
<u></u>
SO lets make e 5 and a 1. 
1, b, 4, d, 5
There has to be more then 1 4 so lets put that as b. 
We can solve for the last remaining digits. 
1+ 4 + 4 + d + 5 / 5 = 4
14 + d /5 = 4
2.2 + d = 4
1.8 = d 
So now if we put in order and replace b with 1.8 and make d as the previous “b” as 4. 
1, 1.8, 4, 4, 5
Thats your 5 numbers right there. 
 
Check:
Mode is 4: yes!
Range is 4: 5-1 = 4 yes!
Median is 4: yes!
Mean is 4: 1 + 1.8 + 4 + 4 +5 / 5 = 4 yes!
Every thing checks out. 
There could be a lot of possibilities. 
For example take this wrong one
Lets make the same exact thing except change the a and the e. 
THis is what we have, 
a, b, 4, d, e
Lets make e and a as 6 and 2.  6-2 is still 4 so its possible. 
2, b, 4, d, 6 
And of course we need more then 1 4 so lets make d 4. 
2, b, 4, 4, 6
Now solve for b in the mean. 
2 + b + 4 + 4 + 6  / 5 = 4
16 +b /5 = 4
3.2 + b = 4 
Solve 
B = 0.8 
This doesn’t work cause the median and the range has constraint here… 
When doing a median, it has to be in ORDER. 
2, 0.8, 4, 4 , 6 isn’t in order
ANd even when put in order.
0.8, 2, 4, 4, 6
THe range has the constraint here becuase 6 - 0.8 isn’t 4.