Yes you can factorise by grouping...

$\displaystyle \begin{align*} x^3 + x^2 + x + 1 &= 0 \\ x^3 + x + x^2 + 1 &= 0 \\ x\left( x^2 + 1 \right) + 1 \left( x^2 + 1 \right) &= 0 \\ \left( x^2 + 1 \right) \left( x + 1 \right) &= 0 \\ x^2 + 1 = 0 \textrm{ or } x + 1 &= 0 \end{align*}$

The first equation doesn't have any real solutions (if you want complex solutions they're $\displaystyle \begin{align*} x = \pm \mathrm{i }\end{align*}$) and the second equation has solution x = -1.