<s>[INST] How do you factor this? $x^3 + x - 2$ How do you factor $x^3 + x - 2$?
Hint: Write it as $(x^3-x^2+x^2-x+2x-2)$ to get $(x-1)(x^2+x+2)$
Note the factored form here. Thanks!
 [/INST] By inspection, we see that $1$ is a root of $x^3 + x - 2$, so it is divisible by $x - 1$; alternatively, the rational roots theorem would suggest this too.
Now $x^2 + x + 2 = x^2 + x + \frac{1}{4} + \frac{7}{4} = (x + \frac{1}{2})^2 + \frac{7}{4}$ has no real roots, and is irreducible. If you're factoring over $\Bbb{C}$, then it's got roots at $\pm \sqrt{\frac 7 4}i - \frac{1}{2}$; denoting these as $\alpha_+$ and $\alpha_-$, the original polynomial then splits as
$$(x - 1)(x - \alpha_+)(x - \alpha_-)$$
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