What are the limitations to adding wings with engines to a helicopter? I've probably seen this in science fiction and cartoons but never in real life. Rotor wing aircraft (helicopters) have numerous advantages over their fixed wing counterparts, and vice versa. For example, helicopters can take off and land vertically and do this almost anywhere. But fixed wing aircraft can fly further, faster and are more fuel efficient than helicopters of similar weight. 

What are the limitations to adding folding wings to a helicopter, each equipped with a turboprop or turbofan whose shafts connects with the helicopter fan blade for vertical take off? 

Once enough speed is gained from the vertical takeoff helicopter phase, the gearbox connects the turboprop or turbofan, they power it enough for it to generate lift at the fixed wings and when the wings are generating lift, the gear box disconnects the helicopter blades and they fold and are housed in the fuselage (like landing gear) to reduce drag. The fixed wing part of the flight takes over for faster speed, more efficient cruise and better range. The same process applies for landing. 

I know there are turbofan aircraft that are a cross, their advantages of fixed wing over rotorwing are limited and their wings do not fold.
A very promising British design from the 1950s used both, a rotor at low speed and a conventional wing at high speed for lift creation. This was the Fairey Rotodyne, a prototype for a flying bus to connect remote locations with a VTOL aircraft. Its rotor was tip-jet-powered and burned a mixture of fuel and compressed air bled from two wing-mounted Napier Eland turboprops. After take-off, power was switched to the propellers and the rotor was unloaded to minimize its drag at high speed. This way, both lift-creating means could be optimized for their speed domain.

You will notice that the wing is unusually  small while the rotor is unusually large. This allowed both to operate at higher efficiency than wings or rotors which are designed to work over the full speed range. However, the combination of the two weighs more than either a rotor or a wing, which reduced the possible payload of the Rotodyne. Neither was meant to fold away because that would had increased weight even more. 



Fairey Rotodyne (picture source Ed Coates collection)

In the end, it was wavering British government support rather than any technical issues which limited the Rotodyne to a single prototype.

Today, the most promising solution seems to be a coaxial rotor with a separate source of thrust; either a propeller or jet engines. The coaxial rotor is slowed down at high flight speed in order to keep tip Mach numbers below 0.9 and the lift asymmetry of conventional helicopters is no problem for a coaxial rotor design, allowing much higher flight speeds.

Folding a wing would not help much: When the wing is not needed (which is at low speed), its friction drag is low. Folding the rotor, on the other hand, makes more sense but so far is limited to a helicopter on the ground in order to reduce its storage space on aircraft carriers. Folding in flight has only been studied conceptually.



Illustration 1B from US patent application 2015/0474290 A1 by Bell Helicopter Textron Inc.