Solve the equation \(x^4+x^3+x^2+x+1=0\).

Multiply by \((x-1)\): \((x-1)(x^4+x^3+x^2+x+1)=x^5-1=0\), with \(x\ne1\).

So the required roots are the fifth roots of unity other than 1: \(x^5=1=\cos2k\pi+i\sin2k\pi\Rightarrow x=\cos\frac{2k\pi}{5}+i\sin\frac{2k\pi}{5}\).

\(k=0\) gives \(x=1\) (rejected). For \(k=1,2,3,4\):

\(x=\cos\frac{2\pi}{5}\pm i\sin\frac{2\pi}{5}=0.3090\pm0.9511i\)
\(x=\cos\frac{4\pi}{5}\pm i\sin\frac{4\pi}{5}=-0.8090\pm0.5878i\)

Roots: \(\cos\frac{2\pi}{5}\pm i\sin\frac{2\pi}{5},\ \cos\frac{4\pi}{5}\pm i\sin\frac{4\pi}{5}\).