Step 1: Place barium.
Barium sits in group 2, the alkaline earth metals. Their nitrates all break down in the same pattern when heated.
Step 2: Write the general decomposition.
For any group 2 nitrate the heated reaction is:
\[ 2M(NO_3)_2 \xrightarrow{\Delta} 2MO + 4NO_2 + O_2 \]
Step 3: Put barium into it.
Replacing M with barium gives:
\[ 2Ba(NO_3)_2 \xrightarrow{\Delta} 2BaO + 4NO_2 + O_2 \]
So we get the oxide, brown nitrogen dioxide fumes and oxygen gas.
Step 4: Match with the options.
The products are $BaO$, $NO_2$ and $O_2$ together.
\[ \boxed{BaO + NO_2 + O_2} \]