let factor-4 = f(a,b,c).
then f(0,b,c)= -(b+c), f(a,0,c)= -(c+a), f(a,b,0)= -(a+b).
so i anticipate that factor-4 = f(a,b,c)=(a+b+c). this also suggested by the fact that factor-4 cannot have second order terms.
although this isn't a sure shot way to say anything about factor-4 but still it must have helped.
another way could be to take partial derivatives w.r.t a, b, c and then compare...
did this help??