Polynomials that share at least one root

The locus of real polynomials $p(x)$ sharing a root with $P(x)$ is the union of the hyperplanes $H_\alpha : p(\alpha)=0$, where $\alpha$ runs over the roots of $P(x)$. This is an arrangement of hyperplanes.

The intersection of $H_\alpha$ with the discriminant variety $D : \mathrm{disc}(p)=0$ contains the subspace $H^{(1)}_{\alpha} : p(\alpha)=p'(\alpha)=0$. In fact $H_\alpha$ and $D$ are tangent along $H^{(1)}_\alpha$. To see this, take a generic polynomial $p_0(x)=(x-\alpha)^2 q_0(x)$ in $H^{(1)}_\alpha$. Near $p_0$, polynomials in $D$ will be of the form $p(x)=(x-\alpha+\varepsilon)^2 q(x)$ where $q$ is near $q_0$. Then $p(\alpha)=\varepsilon^2 q(\alpha)$, while the distance from $p_0$ to $p$ is of first order with respect to $\varepsilon$.

I tried to make a 3D image for $P(x)=x^3+3 x^2-2 x-1$. The set consists of three planes, each tangent to the discriminant surface. But it became too visually complex, partly because the discriminant is complicated. For what it's worth:

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          ^{Discriminant surface: blue.}

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Perhaps someone can do better...