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!

Question

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!

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