Find the maximum and minimum value of $P =x+y+z+xy+yz+zx$

You may use the same method to find the minimum. First, we obtain a lower bound for $P$: $$ \begin{align} P&=x+y+z+xy+yz+zx\\ &=\frac12 [ (x+y+z+1)^2 - (x^2+y^2+z^2) - 1 ]\\ &\ge\frac12 (0 - 27 - 1)\tag{1}\\ &= -14. \end{align} $$ Next, note that at $\left(\frac{\sqrt{53}-1}2,-\frac{\sqrt{53}+1}2,0\right)$, we have $x+y+z+1=0$ and $x^2+y^2+z^2=27$. Hence tie can occur in $(1)$ and the lower bound $-14$ is attainable.

A minimum is $-14$.

Prove that $$(x+y+z)\sqrt{\frac{x^2+y^2+z^2}{27}}+xy+xz+yz+\frac{14}{27}(x^2+y^2+z^2)\geq0$$