If $p, q$ and $r$ are positive rational numbers such that $p>q>r$ and the quadratic $(p+q-2r)x^2+(q+r-2p)x+(r+p-2q)=0$ has a root in $(-1, 0)$, then which of the following statements hold good?

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 $x$