I am asking this question again, because the last response I was given just gave that the the other polynomial was 2x+3, and provided no explanation as to how that was arrived at. Show that 4x 2 +6x+3...

I am asking this question again, because the last response I was given just gave that the the other polynomial was 2x+3, and provided no explanation as to how that was arrived at.

Show that 4x²+6x+3 is a unit in Z₈[x].

All that I know that I have to do is find a polynomial such that

(4x²+6x+3)(polynomial)=1, but I'm not sure the steps in general to follow. I think after multiplying the two polynomials the coefficients will have to turn out to be 0 mod 8 and the constant term will have to be 1 mod 8.  How do I approach any problem of this type?  I have no examples to go by, and I need to know the thinking process that is needed to solve these types of problems.  Thanks!