plotting - Understand MeshFunctions with Intercepts

Bug introduced in 7.0.1 or earlier and persisting through 11.0.1 or later

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While I've been thinking about this question, I tried to visualize the Intercepts. I use three snippets, actually snippet 1, 2 and 3. The visualization works as expected.

p[x_] := x^4 - 4 x^2 - 2 x + 1

solutions = NSolve[p[x] == 0, x]

snippet1 = 
  Plot[p[x], {x, -3, 3}, PlotStyle -> Green, 
   MeshFunctions -> {p[x] /. x -> # &}, Mesh -> {{0}}, 
   MeshStyle -> Directive[PointSize[Large], Red], 
   PlotLabel -> "Snippet 1"];


snippet2 = 
  Show[Plot[p[x], {x, -3, 3}, PlotStyle -> Brown], 
   Graphics[{Red, PointSize[Large], Point[{x, p[x]} /. solutions]}], 
   PlotLabel -> "Snippet 2"];

snippet3 = 
  Plot[p[x], {x, -3, 3}, PlotStyle -> Blue, 
   Epilog -> {PointSize[Large], Red, Point[{x, 0} /. NSolve[p[x]]]}, 
   PlotLabel -> "Snippet 3"];

When applying snippet 1 with a larger xmin-value a fifth red spot appears. I do not observe this behavior on snippet 2 and 3. What's wrong with snippet 1, or with my knowledge/interpretation?

snippet1 = 
 Plot[p[x], {x, -3, 4}, PlotStyle -> Green, 
  MeshFunctions -> {p[x] /. x -> # &}, Mesh -> {{0}}, 
  MeshStyle -> Directive[PointSize[Large], Red], 
  PlotLabel -> "Snippet 1"]

I'm on 10.0 for Mac OS X x86 (64-bit) (September 10, 2014).

Answer

A very good question. You should not rely on MeshFunctions.

Follow the old path step for step:

fun = x^4 - 4 x^2 - 2 x + 1;

xvals = NSolve[fun == 0, x] // Values // Flatten

{-1.48119, -1., 0.311108, 2.17009}

yvals = fun /. xvals // Chop

{0, 0, 0, 0}

zeroes = Transpose[{xvals, yvals}]

{{-1.48119, 0}, {-1., 0}, {0.311108, 0}, {2.17009, 0}}

snippet1 =
 Plot[fun, {x, -3, 5},
  Epilog -> {Red, PointSize[0.02], Point /@ zeroes},
  PlotStyle -> Green,
  PlotLabel -> "Snippet 1"]