If $a,b,c \in \mathbb{R}$ and $a+b+c = 1$, prove that $(2a+b)(2b+c)(2c+a)+(1+a+2b)(1+b+2c)(1+c+2a) \leq 9$ If $a,b,c \in \mathbb{R}$ and $a+b+c = 1$, prove that
$(2a+b)(2b+c)(2c+a)+(1+a+2b)(1+b+2c)(1+c+2a) \leq 9$

Hint :
As  pointed out by user Albus Dumbledore we can just consider the case where all the variable are positive now try to show that with this constraint of positivity :
$$(2a+b)(2b+c)(2c+a)\leq 1$$
And :
$$(1+a+2b)(1+b+2c)(1+c+2a)\leq 8$$
