Factorization of $x^5+x$ I need to make the decomposition in $\mathbb{R}$ of:
$$x^5+x$$
Here my steps:
$$x(x^4+1)$$
$$x\big[ (x^2)^2+1 \big]$$
$$x\big[(x^2+1)^2-2x^2\big]$$
how should I proceed?

Another (less ingenious but more systematic?) way is to first work over complex numbers:
$$\begin{align}x^4+1&=(x^2)^2-i^2=(x^2-i)(x^2+i)\\
&=(x+\frac{1+i}{\sqrt2})(x-\frac{1+i}{\sqrt2})\cdot(x+\frac{-1+i}{\sqrt2})(x-\frac{-1+i}{\sqrt2})\\
&=(x-\frac{1+i}{\sqrt2})(x+\frac{-1+i}{\sqrt2})\cdot(x-\frac{-1+i}{\sqrt2})(x+\frac{1+i}{\sqrt2})\\
&=(x^2-\sqrt2x+1)(x^2+\sqrt2x+1)\end{align}$$
So this obtains the decomposition by permuting the two hidden factors over complex numbers.
Hope this helps.
