Proving $x^{4}+x^{3}+x^{2}+x+1$ is always positive for real $x$

For $n$ odd:

• if $x\neq 1$: $$ 1+x+\cdots+x^{n-1} = \frac{x^n-1}{x-1}\neq 0 $$
• if $x=1$: $$1+x+\cdots+x^{n-1}=n> 0$$

As it is a continuous function the intermediate value theorem concludes that it is $>0$.

It turns out that any polynomial or rational function that is always positive can be written as a sum of squares.

e.g.

$$ x^4 + x^3 + x^2 + x + 1 = \left(\frac{x^2 + x}{\sqrt{2}}\right)^2 + \left(\frac{x + 1}{\sqrt{2}}\right)^2 + \left(\frac{x^2}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 $$

Alas, I don't know of any systematic way to figure out how to come up with such a representation, although this one is easily extended to the particular family of polynomials you are interested in.