If $q\ne 0$ and the equation $x^3+p{x}^2+q=0$ has a root of multiplicity 2, then $\text{'}p\text{'}$ and $\text{'}q\text{'}$ are connected by

Let $f\left(x\right)=x^3+p{x}^2+q=0$

Since $f\left(x\right)=0$ has a root of multiplicity 2

$f\left(x\right)=0$ and $f\text{'}\left(x\right)=3x^2+2px=0$ have a common root

The roots of $3x^2+2px=0$ and $x=0$ and $x=\frac{-2p}{3}$

But $x=0$ is not a root of $f\left(x\right)=0$

Common root of $f\left(x\right)=0$ and $f\text{'}\left(x\right)=0$ is $x=\frac{-2p}{3}$

${\left(\frac{-2p}{3}\right)}^3+p{\left(\frac{-2p}{3}\right)}^2+q=0$

$4p^3+27q=0$.