= Solution
Put $Y=X^2$. Modulo two, $Y^2+9Y-2$ has the two simple roots zero and one, so <Hensel lemma> lifts them to roots $y_0,y_1\in\mathbb Z_2$, with $v_2(y_0)=1$ and $y_1\equiv1\pmod8$. The <2-adic unit> criterion makes $y_1$ a square, so $X^2-y_1$ splits into two linear factors. The polynomial $X^2-y_0$ is Eisenstein and remains irreducible. Thus $X^4+9X^2-2$ has three irreducible factors, of degrees $1,1,2$, as in <factorization of X4 plus 9X2 minus 2 over the 2-adic numbers>.
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