# Show that $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$

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

• And your question is? – Eevee Trainer Apr 2 at 22:21
• Is this right?? – Dillain Smith Apr 2 at 22:21

Except that you probably meant to write mod $$8$$ at the end, rather than mod $$9$$, this looks good.