Problem

A function $f$ is defined by $f(x)=px^2+4x+q$ for $x\in\mathbb{R}$, where $p$ and $q$ are constants.

(a) It is given that $p=2$ and $q=10$.

(i) Express $f(x)$ in the form $a(x+b)^2+c$, where $a$, $b$ and $c$ are constants.

(ii) State the range of $f$.

(b) It is given instead that $q=-5$ and the roots of $f(x)=0$ are $5m$ and $-9m$, where $m$ is a constant.

Find the values of $p$ and $m$.