The condition that the roots of $x^3+3p{x}^2+3qx+r=0$ may be in A.P is

Let the roots be a-d, a, a+d
$S_1=a-d+a+a+d=-3p\Rightarrow a=-p$
a  is a root of $x^3+3p{x}^2+3qx+r=0$ then 
$\begin{array}{l}\Rightarrow a^3+3p{a}^2+3qa+r=0\\ \Rightarrow (-p{)}^3+3p(-p{)}^2+3q(-p)+r=0\\ \Rightarrow 2p^3+r=3pq\end{array}$