Final answer:
The advantage of a heat pump over burning fuel is that it transfers heat from the outside air to the indoors, providing additional heat at no cost. When burning fuel, you pay for all of it. However, the work input for a heat pump is sometimes more expensive than burning fuel.
Step-by-step explanation:
The great advantage of using a heat pump to keep your home warm, rather than just burning fuel, is that a heat pump supplies Qc + W.
Heat transfer is from the outside air, even at a temperature below freezing, to the indoor space. You only pay for W, and you get an additional heat transfer of Qc from the outside at no cost; in many cases, at least twice as much energy is transferred to the heated space as is used to run the heat pump.
When you burn fuel to keep warm, you pay for all of it. The disadvantage is that the work input (required by the second law of thermodynamics) is sometimes more expensive than simply burning fuel, especially if the work is done by electrical energy.