While adding it is to be noted that in \(\displaystyle{Z}_{{8}}\) the coefficients can take values from 0,1,2,3,4,5,6,7. And after adding the coefficients resulting term is taken modulo 8. For example 7+8=15 but this is not in \(\displaystyle{Z}_{{8}}\) so taking it modulo 8 gives 7, that is, 7+8=7 modulo 8. So in \(\displaystyle{Z}_{{8}}\),7+8=7.

Given polynomials to be added are \(\displaystyle{g{{\left({x}\right)}}}={4}{x}^{{2}}+{4}{x}+{6},{h}{\left({x}\right)}={6}{x}^{{2}}+{3}\). To add them, add the coefficients of like terms and then apply modulo 8.

\(\displaystyle{g{{\left({x}\right)}}}+{h}{\left({x}\right)}={\left({4}{x}^{{2}}+{4}{x}+{6}\right)}+{\left({6}{x}^{{2}}+{3}\right)}\)

\(\displaystyle={\left({4}{x}^{{2}}+{6}{x}^{{2}}\right)}+{4}{x}+{\left({6}+{3}\right)}\)

\(\displaystyle={10}{x}^{{2}}+{4}{x}+{9}\)

\(\displaystyle={2}{x}^{{2}}+{4}{x}+{1}\) modulo 8

Hence, \(\displaystyle{g{{\left({x}\right)}}}+{h}{\left({x}\right)}={2}{x}^{{2}}+{4}{x}+{1}\)