A point $(a,b)$ on $y^2 - ax^3 - bx^2 -cx - d$ is a simple point if and only if $\frac{\partial}{\partial x}(y^2-p(x))\Bigm|_{(x,y)=(a,b)}\neq 0$ or $\frac{\partial}{\partial y}(y^2-p(x))\Bigm|_{(x,y)=(a,b)} \neq 0$. It is a multiple point if and only if both partial derivatives are zero.
Suppose that $(a,b)$ is not a simple point. Then $\begin{align*} \frac{\partial}{\partial x}(y^2-p(x))\Bigm|_{(x,y)=(a,b)} = p'(a) &=0\\ \frac{\partial}{\partial y}(y^2-p(x))\Bigm|_{(x,y)=(a,b)} = 2b&=0. \end{align*}$ From the second equation, we have that $b=0$. Hence, $0^2 = p(a)$, so $a$ is a root of both $p(x)$ and $p'(x)$, hence $a$ is a multiple root of $p(x)$. This proves that if $(a,b)$ is a multiple point, then $b=0$ and $a$ is a multiple root of $p(x)$.