suppose the width of the path is x.
the dimensions of the extended rectangle are (12+x+x), and (16+x+x)
(12+2x)(16+2x)=285
4x²+56x+192=285
4x²+56x-93=0
Not easy to factor it, so use the quadratic formula to solve:
x={-56+√[(56²-4*4*(-93)]}/2*4 or x={-56-√[(56²-4*4*(-93)]}/2*4
x=1.5 (ignore the negative solution.)