a) To find: The factor of the polynomial into linear and irreducible quadratic factor with real coefficients.

Consider, \(\displaystyle{P}{\left({x}\right)}={x}^{{{4}}}+{8}{x}^{{{2}}}+{16}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{4}}}+{4}{x}^{{{2}}}+{4}{x}^{{{2}}}+{16}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{2}}}{\left({x}^{{{2}}}+{4}\right)}+{4}{\left({x}^{{{2}}}+{4}\right)}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={\left({x}^{{{2}}}+{4}\right)}{\left({x}^{{{2}}}+{4}\right)}\)

Therefore, \(\displaystyle{P}{\left({x}\right)}={\left({x}^{{{2}}}+{4}\right)}{\left({x}^{{{2}}}+{4}\right)}\).

b) To find: The factor completely into linear factors with complex coefficients.

Consider, \(\displaystyle{P}{\left({x}\right)}={x}^{{{4}}}+{8}{x}^{{{2}}}+{16}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{4}}}+{4}{x}^{{{2}}}+{4}{x}^{{{2}}}+{16}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{2}}}{\left({x}^{{{2}}}+{4}\right)}+{4}{\left({x}^{{{2}}}+{4}\right)}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={\left({x}^{{{2}}}+{4}\right)}{\left({x}^{{{2}}}+{4}\right)}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={\left({x}-{2}{i}\right)}{\left({x}-{2}{i}\right)}{\left({x}+{2}{i}\right)}{\left({x}-{2}{i}\right)}\)

Therefore, \(\displaystyle{P}{\left({x}\right)}={\left({x}-{2}{i}\right)}{\left({x}-{2}{i}\right)}{\left({x}+{2}{i}\right)}{\left({x}-{2}{i}\right)}\).

Consider, \(\displaystyle{P}{\left({x}\right)}={x}^{{{4}}}+{8}{x}^{{{2}}}+{16}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{4}}}+{4}{x}^{{{2}}}+{4}{x}^{{{2}}}+{16}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{2}}}{\left({x}^{{{2}}}+{4}\right)}+{4}{\left({x}^{{{2}}}+{4}\right)}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={\left({x}^{{{2}}}+{4}\right)}{\left({x}^{{{2}}}+{4}\right)}\)

Therefore, \(\displaystyle{P}{\left({x}\right)}={\left({x}^{{{2}}}+{4}\right)}{\left({x}^{{{2}}}+{4}\right)}\).

b) To find: The factor completely into linear factors with complex coefficients.

Consider, \(\displaystyle{P}{\left({x}\right)}={x}^{{{4}}}+{8}{x}^{{{2}}}+{16}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{4}}}+{4}{x}^{{{2}}}+{4}{x}^{{{2}}}+{16}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={x}^{{{2}}}{\left({x}^{{{2}}}+{4}\right)}+{4}{\left({x}^{{{2}}}+{4}\right)}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={\left({x}^{{{2}}}+{4}\right)}{\left({x}^{{{2}}}+{4}\right)}\)

\(\displaystyle\Rightarrow{P}{\left({x}\right)}={\left({x}-{2}{i}\right)}{\left({x}-{2}{i}\right)}{\left({x}+{2}{i}\right)}{\left({x}-{2}{i}\right)}\)

Therefore, \(\displaystyle{P}{\left({x}\right)}={\left({x}-{2}{i}\right)}{\left({x}-{2}{i}\right)}{\left({x}+{2}{i}\right)}{\left({x}-{2}{i}\right)}\).