In completion to @JulianHzg's anwser:
Increasing the bypass ratio is very desirable in order to improve the efficiency of the engine, and for the same amount of thrust, this means increasing the fan diameter
Increasing the bypass ratio is surely a way to reach a higher efficiency (and maybe the most developed topic at the moment in aircraft engines), but only if you can provide the engine with a powerful enough gas generator, so increasing the bypass ratio is constrained by other parameters such as combustion chamber temperature and turbine efficiency.
In order to keep the fan mostly subsonic, the fan rotational speed has to decrease with increasing fan diameters. Supersonic fan blade tips are possible and in use, but you lose efficiency and increase noise, so it's desirable to limit this to the outermost area.
Also, blades permanently submitted to shock waves tend to have a very short lifespan because of very high cycle fatigue. Therefore, people tend to avoid supersonic flow conditions.
For the same efficiency, a low-speed turbine requires more stages and therefore more weight compared to a high-speed turbine. Compare the PW1100G's 3-stage LPT (geared) with LEAP-1A's 7-stage LPT (not geared).
Basically, the rotational speed of every element must be adapted to the fluid velocity. So if every rotating element is driven by the same shaft, the flow must be adapted adding stages in the turbine or compressor modules. If you have two shafts, one slow (good for the fan) and one fast (good for the turbine) you get a better compromise in terms of efficiency. However, this poses other issues (rotordynamic instabilities for instance) which explains why Rolls-Royce is the only manufacturer to propose three shafts (even better in terms of efficiency but also very complex).
The gearbox allows to keep using two shafts while improving adaptation between fluid and rotating element.
The question about the weight of the gearbox is very important, is must be recalled that a gearbox mut be able to transmit the power of the LPT to the fan, so it's not a simple gearbox from your car. Also, although epicycloïdal gearboxes have a very good efficieny transmitting power using such trains costs energy, creates heat, requires lubrication and maintenance.
To conclude, this is a very well know technology that is even well known to the aeronautic industry (how do you think helicopters fly !), but that was never used on high-power aircraft engines. Another application of gearboxes are the open-rotor engines which involve even more complex transmissions du to counter rotating fans.