Step 1: Identify the coordination entity.
The given complex is
\[
[Co(NH_2CH_2CH_2NH_2)_2Br_2]Br
\]
The coordination sphere is
\[
[Co(NH_2CH_2CH_2NH_2)_2Br_2]^+
\]
and the counter ion outside the coordination sphere is
\[
Br^-
\]
Step 2: Identify the ligands.
The ligand
\[
NH_2CH_2CH_2NH_2
\]
is ethylene diamine, commonly written as
\[
en
\]
It is a neutral bidentate ligand.
There are two ethylene diamine ligands, so the prefix used is
\[
bis
\]
because ethylene diamine is a polydentate ligand.
There are also two bromide ligands inside the coordination sphere. These are named as
\[
dibromido
\]
Step 3: Find the oxidation state of cobalt.
Let the oxidation state of cobalt be
\[
x
\]
Ethylene diamine is neutral, so its charge is
\[
0
\]
Each bromido ligand has charge
\[
-1
\]
There are two bromido ligands, so total charge due to bromido ligands is
\[
-2
\]
The complex ion has charge
\[
+1
\]
Therefore,
\[
x+0-2=+1
\]
\[
x=+3
\]
Thus, cobalt is in the
\[
+3
\]
oxidation state.
Step 4: Write the IUPAC name.
Ligands are named before the metal.
The anionic ligand bromido is written as
\[
dibromido
\]
The neutral ligand ethylene diamine is written as
\[
bis(ethylene diamine)
\]
The metal is cobalt with oxidation state
\[
(III)
\]
The counter ion is bromide.
Therefore, the name is
Dibromido bis(ethylene diamine) cobalt (III) bromide
Step 5: Final conclusion.
Hence, the IUPAC name of the complex is
\[
\boxed{\text{Dibromido bis(ethylene diamine) cobalt (III) bromide}}
\]
Therefore, the correct option is
\[
\boxed{(4)}
\]