Let $a+b+c=4$  and $a^2+b^2+c^2+3(ab+bc+ac)=21$  where $a,b,c\in R^{+}$ , then the maximum value of $(a+b)(b+c)(c+a)$  is

$(a+b)+(b+c)+(c+a)=8$

$\Rightarrow$  $\alpha +\beta +\gamma =8$  where $\alpha =a+b$ , $\beta =b+c$ , $\gamma =c+a$

And $a^2+b^2+c^2+3\left(ab+bc+ca\right)=21$

$\alpha$ , $\beta$ , $\gamma$  are real roots of equation $x^3-8x^2+21x-\lambda =0$  

where $\lambda =(a+b)(b+c)(c+a)$

Let $f\left(x\right)=x^3-8x^2+21x-\lambda$

$\Rightarrow$  $f\text{'}\left(x\right)=3x^2-16x+21=0$

$\Rightarrow$  $x=3,\frac{7}{3}$

$\Rightarrow$  $\frac{490}{27}\le \lambda \le 18$