Find the factorization of the polynomial as a product of irreducible on rings R[x] and C[x]

Question

Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $\Bbb R[x]$ and $\Bbb C[x]$

Testing with the simplest possible root in this case, $P(1) = 0$

Applying the schema of Ruffini

$ \begin{array}{c|lcr} & 1 & -1 & 8 & -8 & 16 & -16 \\ 1 & & 1 & 0 & 8 & 0 & 16 \\ \hline & 1 & 0 & 8 &0 &16 & 0 \\ \end{array} $

$P(x)=(x-1).(x^4+8x^2+16)$

So, I need to make it irreducible in $R[x]$, there's no more roots, but I don't know how to reduce it, I tried factoring: $x^4+8.(x^2+2).(x-1)$, but I can not keep working it.

What I can do to solve it?

Answer 1

Remember $a^2+2ab+b^2=(a+b)^2$. Thus $x^4+8x^2+16=\,?$ Any polynomial $x^2+a$ with the number $a>0$ is irreducible over $\bf R$, but it should be clear how to reduce them over $\bf C$.