How can I prove that $27\leq \left( x^{2}+2\right) \left( y^{2}+2\right) \left( z^{2}+2\right) \leq 44,\ x+y+z=3$

Question

If $x,y,z\in [0,\infty )$ such that $x+y+z=3$ prove that $27\leq \left( x^{2}+2\right) \left( y^{2}+2\right) \left( z^{2}+2\right) \leq 44$. I tried to use the relation $x^{2}+2\leq x^{2}+3x+2,x\geq 0$, but without success.