If the quadratic equation $a{x}^2-bx+12=0,$ where $a$ and $b$ are positive integers not exceeding $10,$ has roots both greater than $2,$ then the number of possible ordered pair $\left(a, b\right),$ is

Let $p\left(x\right)=x^2+ax+b$ have two distinct real roots, where $a,b$ are real number. Define $g\left(x\right)=p\left(x^3\right)$ for all real number $x$

Then, which of the following statements are true?
I. $g$ has exactly two distinct real roots.
II. $g$ can have more than two distinct real roots.
III. There exists a real number $\alpha$ such that $g\left(x\right)\ge \alpha$ for all real $x$