$Q\le \prod \frac{5+2x}{1+x}\le P$ find $P,Q$

For $x=y=z=\frac{1}{3}$ we obtain a value $\frac{4913}{64}.$

We'll prove that it's a minimal value.

Indeed, after homogenization we need to prove that $$\prod_{cyc}\frac{7x+5y+5z}{2x+y+z}\geq\frac{4913}{64}$$ or $$\sum_{sym}(687x^3+489x^2y-1176xyz)\geq0,$$ which is true by AM-GM or by Muirhead.

Also, for $y=z\rightarrow0^+$ we obtain a value $\frac{175}{2}.$

We'll prove that it's a supremum of the expression.

Indeed, we need to prove that: $$\prod_{cyc}\frac{7x+5y+5z}{2x+y+z}\leq\frac{175}{2}$$ or $$\sum_{sym}(135x^2y+94xyz)\geq0,$$ which is obvious.