My attempt is: We guess for $p(x)$ an inverse polynomial $q(x)=2x+3$,
$(4x^2+6x+3)(2x+3)=8x^3+12x^2+6x+12x^2+18x+9=1\pmod{8}$.
The existence of such an inverse verifies this is a unit in $\mathbb{Z}_8[x]$
Edit:Typo
2026-04-07 00:21:37.1775521297
Show that $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$
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Except that you probably meant to write mod $8$ at the end, rather than mod $9$, this looks good.