Step 1: Know that nitrates decompose on heating.
When metal nitrates are heated, they break down. The products depend on the metal. We must find the observation that is the same for both lithium nitrate and calcium nitrate.
Step 2: Heat lithium nitrate.
Lithium nitrate behaves unlike other alkali metal nitrates. It breaks into the oxide, brown nitrogen dioxide, and oxygen: \[ 4LiNO_3 \rightarrow 2Li_2O + 4NO_2 + O_2. \]
Step 3: Heat calcium nitrate.
Calcium nitrate breaks into the oxide, nitrogen dioxide, and oxygen too: \[ 2Ca(NO_3)_2 \rightarrow 2CaO + 4NO_2 + O_2. \]
Step 4: Look at the gases.
Both reactions give off $NO_2$, which appears as brown fumes. Both also give off $O_2$, which supports combustion (it relights a glowing splint). So both show the same two gas clues.
Step 5: Reject the wrong options.
The gas is not colorless, so the colorless gas option is wrong. No free metal is produced, so that option is wrong. Gases clearly are released, so the no gas option is wrong.
Step 6: State the common observation.
The matching observation for both is \[ \boxed{\text{brown fumes and a gas that supports combustion}} \]