If $a$, $b$, $c$ are sides of a triangle, prove $2(a+b+c)(a^2+b^2+c^2)\geq3(a^3+b^3+c^3+3abc)$ 
$a$, $b$, $c$ are sides of a triangle, prove:
  $$2(a+b+c)(a^2+b^2+c^2)\geq3(a^3+b^3+c^3+3abc)$$

What I have tried:
$$
⇔2\sum (a+b)ab\geq \sum a^3+9abc
$$
so I can't use 
$$\sum a^3+3abc\geq \sum ab(a+b)$$

WLOG $$c=\max\{a,b,c\}\implies a=x+u,b=x+v\text{  and  }c=x+u+v (x>0, u,v\ge 0)$$ 
Then your inequality is equivalent to
$$x(u^2-uv+v^2)+2(u+v)(u-v)^2\ge 0*\text{true}*$$
