Do the four engines of a BAe-146 make it efficient?
Do the four engines make the BAe efficient? No! Does it affect performance? Yes it does. The BAe 146 occupied a niche for Hot &amp; High airfields, in that it can continue the takeoff when an engine fails: one failed engine out of four leaves 75% of full thrust. So take-off performance with a failed engine is better on a 4-engine aeroplane. 

The engines themselves were quite clever designs: the Lycoming ALF 502 with a relatively high bypass ratio and a geared fan. The bypass ratio of 5.7:1 is higher than the 5.0:1 of the Trent 700 used for the A330. By comparison, the Fokker 100 used the Rolls Royce Tay, with a bypass ratio of 3.0:1 and the proven core of the RR Spey.

 Photo from the wiki.

But was the BAe 146 efficient because of the four engines? For an aircraft of a given weight and a given thrust requirement, the design considerations for number of engines would be:


Four engines are each smaller than two engines. Turbine engines do not scale down well. All else being equal, purchase price starts to be the limiting factor when reducing the size: half thrust engines are not half the price. 
4-engined planes have worse fuel economy than an identical sized twin engine, due to decreased efficiency of a smaller turbine, and increased drag from more engine pods and a larger vertical tail area. 
In cruise, a typical single aisle airliner uses a third of the take-off and climb thrust. So once in cruise, one out of two engines can provide all the thrust required to continue the flight, and have less yawing moment than a failed outboard engine of four. Picture and quote from this document: 





  Specifically, current subsonic airliners require about three to five times more thrust to take off than they do to cruise, and the power produced by a high bypass ratio turbofan engine at constant throttle setting varies in much the same way.


Twin engine turbofans actually have a third engine on board: the APU. It can power the systems with full redundancy after an engine fail, and is normally shut off in flight, like the no.1 engine of a P-3 when loitering. Note that the propeller blades of the P-3 can be made to feather, providing much less asymmetrical drag than a windmilling turbofan.

So the advantages of four engines are mainly the consequences of an engine fail in the TO/climb phase: a four engined aircraft could opt to continue the flight, while the twin engine would have no more redundancy and a much harder time climbing to altitude. But all else being equal, four engines have twice the chance of failure than two engines...

The BAe 146 had a relatively wide and stubby fuselage, designed for 6-abreast seating but often configured for 5-abreast due to awkward headroom for the window passengers. This added to the drag penalty of the type over that of a comparable size aircraft like the Fokker 100, which is still a sought after aircraft for feeder airlines in Australia, the limiting and most expensive cost being engine overhaul. The BAe 146 has twice this cost...