4
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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

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  • $\begingroup$ And your question is? $\endgroup$ – Eevee Trainer Apr 2 at 22:21
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    $\begingroup$ Is this right?? $\endgroup$ – Dillain Smith Apr 2 at 22:21
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Except that you probably meant to write mod $8$ at the end, rather than mod $9$, this looks good.

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  • $\begingroup$ Yes you are correct its supposed to be mod 8, thanks $\endgroup$ – Dillain Smith Apr 2 at 22:22

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