College Algebra, Chapter 4, 4.5, Section 4.5, Problem 66

(a) Factor $P(x) = x^4 + 8x^2 + 16$ into linear and irreducible quadratic factors with real coefficients. (b) Factor $P$ completely into linear factors with complex coefficients.


$
\begin{equation}
\begin{aligned}
\text{a.) } P(x) &= x^4 + 8x^2 + 16\\
\\
&= (x^2 + 4)^2 && \text{Perfect square}
\end{aligned}
\end{equation}
$

The factor $(x^2 + 4)^2$ is irreducible, since it has no real zeros.

b.) To get the complete factorization, we set $x^2 + 4 = 0$, so $x = \pm 2i$
Thus,

$
\begin{equation}
\begin{aligned}
P (x) &= (x^2 + 4)^2\\
\\
&= [(x-2i)(x+2i)]^2\\
\\
&= (x-2i)^2 (x+2i)^2\\
\\
&= (x-2i)(x-2i)(x+2i)(x+2i)
\end{aligned}
\end{equation}
$