No, you are onto something. I like to use the variable $z$ for this. I forget what happens if you factor $z^3 - 2 \pmod 3.$ After that, there is one simple pattern when $p \equiv 2 \pmod 3,$ when you factor $z^3 - 2 \pmod p.$ That is, a linear factor times a quadratic.
Here is the cute bit: once $p \equiv 1 \pmod 3,$ and you factor $z^3 - 2 \pmod p,$ you get two wildly different outcomes: if there is an expression $p = x^2 + 27 y^2$ in integers, you get three linear terms, distinct. However, if $p = 4 x^2 + 2 x y + 7 y^2,$ irreducible.
A similar example, see Numbers represented by a cubic form
$$ z^3 - z^2 - z - 1 $$ separately for primes with Legendre $(p|11) = -1$ and then for $p = x^2 + 11 y^2$ and then for $p = 3 x^2 + 2 x y + 4 y^2.$ As before, when there is an $xy$ term, you need to allow $xy$ both positive and negative to get all possible such primes. For example, with $x=1,y=-1,3 x^2 + 2 x y + 4 y^2 =5. $
Quartic examples: factor $z^4 + 3 \pmod p,$ when (A) $p=2,3$, (B) larger $p \equiv 3 \pmod 4,$ (C) $p = 5 x^2 \pm 4 xy + 8 y^2,$ (D) $p = 4 x^2 + 9 y^2,$ (E) $p = x^2 + 36 y^2$
Factor $z^4 + 2 z^2 - 7 \pmod p,$ when (A) $(-56|p) = -1,$ (B) $p = 3 x^2 \pm 2 xy + 5 y^2,$ (C) $p = 2 x^2 + 7 y^2,$ (D) $p = x^2 + 14 y^2.$ This is the one that is worked out in full in David A. Cox, Primes of the Form $x^2 + n y^2.$ On page 188 of that book, Theorem 9.12 gives the frequencies of the types of primes I have been specifying. This is an application of the Chebotarev Density that Qiaochu mentions.
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