A friend asked me$:$
For $a,b,c \in [\,1,2\,].$ Find $f(a,b,c)$ and $g(a,b,c)$ where $f(a,b,c),g(a,b,c)$ are all non-negative polynomials such as$:$ $$7abc- ab\left ( a+ b \right )- bc\left ( b+ c \right )- ca\left ( c+ a \right )= f\left ( a, b, c \right )+ \left ( a- b \right )g\left ( a, b, c \right )$$
I tried and I found $$f(a,b,c)={\dfrac {a \left(2a-c \right) \left( 2c-a \right) \left( b-1 \right) ^{2}}{ \left( a-1 \right) ^{2}}}$$ is satisfied (from this we can get $g(a,b,c)$ non-negative also).
Any other$?$ Thanks for a real lot!
Forgive for my bad English!