You might want to look up Surd (https://reference.wolfram.com/language/ref/Surd.html).
Especially, you might want to look at the "Possible Issues" section where it says:
"On the negative real axis, Surd[x,n] is different from the principal root returned by Power
and look a the following examples. If you look at Surd instead of Power then
1 + Surd[-6,3] == 1 - Surd[6,3]
is actually True. Rather than
1 + (-6)^(1/3) == 1 - 6^(1/3)
which is False. Note that
Reduce[Surd[x^3 + 2 x^2 + 3 x, 3] - x - 1 <= 0, x]
gives the solution that you expect if I am not mistaken.
Cheers,
Marco
