What are all of the real roots of the following polynomial? f(x) = x4 - 24x2 - 25

It factors as ...
  f(x) = (x² -25)(x² +1) = (x -5)(x +5)(x² +1)
A root will make one of the factors equal to zero. The quadratic factor has only complex roots.

The real roots are x = ±5.