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Ær,NċAƲƇ

A monadic Link that accepts a list of polynomial coefficients ([...,d,c,b,a]) and yields the real roots.

Try it online! Or see the test-suite.

How?

The heavy lifting is done by a built-in...

Ær,NċAƲƇ - Link: coefficients, P
Ær       - polynomial roots (of P)
       Ƈ - keep those for which:
      Ʋ  -   last four links as a monad - f(x):
   N     -     negate (x)
  ,      -     (P) paired with (that)
     A   -     absolute value (of x)
    ċ    -     count occurrences (of that in the pair) -> 1 if x is real, 0 if not