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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          [![enter image description here][1]][1]
          ^{Discriminant surface: blue.}

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Perhaps someone can do better...