How to celebrate this holiday correctly

$x$ , $y$ , $z$ are three numbers such that $x, y, z \in [2; 4]$ and $x + y + z = 9$ . The minimum value of the following expression occurs when $y = a\sqrt b + c$ with $b$ is square - free.

$\large \dfrac{(x + y + z)^2 - (2x + y + 2z)}{(x - y + z)^2 + (x + 2y + z)}$

Calculate the value of $ab^2 + c$ .