The two "E"s make things a little tricky here. Note that in a rearrangement the "E"s are identical, as swapping them doesn't change anything.
Let's look at a smaller example, were we can list everything. How many different rearrangements of "ROAR" are there that contain "OAR" in the sequence.
Here there's only $2! = 2$ (not $2!\cdot 2 = 4$) different rearrangements that contain "OAR":
- ROAR
- OARR
I think it also helps to list out ALL the different rearrangements of "ROAR" in total (note there are not $4! = 24$ rearrangements, there are $4!\div 2 = 12$ of them:
- AORR
- AROR
- ARRO
- OARR
- ORAR
- ORRA
- RAOR
- RARO
- ROAR
- RORA
- RRAO
- RROA
Hope this helps!