The garden area is 12*16=192 sq m.
The path forms a larger rectangle around the garden rectangle, and its sides have lengths 16+2w and 12+2w, with an area of (16+2w)(12+w) [draw a picture; it will help].
Expanding the brackets: 192+32w+24w+4w^2 [this is where 4x^2 comes in in the other answer]=285, the total area of garden and path. So we can write this: 4w^2+56w-93=0. 93 is the difference between the garden area and the total area with path.
Hopefully this factorises: (2w-3)(2w+31)=0. Obviously w must be positive for our purposes so we take the solution 2w=3 metres, making w=1.5m.
Let's check it out: width of wider garden=15 and length=19. Area=19*15=285 sq m.