optimization – Prove that there exists at least a form $gleft ( a, b, c right )pm fleft ( a, b, c right )left ( a- b right )left ( b- c right )$ of $H$

Given a non-negative cyclic polynomial $H$ with the form $mleft ( a, b, c right )left ( c- 1 right )left ( a- 1 right )+ nleft ( a, b, c right )$ so that
$$mleft ( a, b, c right ), nleft ( a, b, c right )geq 0$$
Prove that there exists at least a form $gleft ( a, b, c right )pm fleft ( a, b, c right )left ( a- b right )left ( b- c right )$ of $H$ so that
$$gleft ( a, b, c right ), fleft ( a, b, c right )geq 0$$

Remark. The inverse problem is not true, e.g for_ A problem related to minimizing the multiplication signs
$$left ( c- a right )^{2}- left ( a- b right )left ( b- c right )= frac{1}{4}left ( 2c+ 2a- b right )^{2}+ frac{3}{4}b^{2}- 3cageq 0$$
There only exists $constant= -3< 0,$ in the problem we see $a, b, cequiv a- 1, b- 1, c- 1$ that makes no difference..

If we can’t come to a general solution, I will give the the best form with least multiplication signs, hope the give the form like the problem as follow :
$$left ( abc+ a+ b+ c right )^{3}- 8abcleft ( 1+ a right )left ( 1+ b right )left ( 1+ c right )geq 0$$
One form given_Given three positive numbers $a,,b,,c$ . Prove that $(!abc+ a+ b+ c!)^{3}geqq 8,abc(!1+ a!)(!1+ b!)(!1+ c!)$ .