Factorization of X4 plus 9X2 minus 2 over the 2-adic numbers ($X^4+9X^2-2$ over ${\mathbb{Q}}_2$)

The polynomial $X^4+9X^2-2$ has three irreducible factors over ${\mathbb{Q}}_2$, of degrees one, one, and two. Its quadratic in $X^2$ has one unit root congruent to one modulo eight, hence a square, and one root of valuation one, whose associated quadratic factor is Eisenstein.