a^3+a^2+a+1=0

Factoring by "grouping,"  we have

a^2(a + 1) + 1 (a +1) = 0       Taking out the common factor, a+1, we have

(a+1)(a^2+1) = 0          Setting the first factor to 0, we have

a+1 = 0        Subtract 1 from both sides

a = -1            Setting the second factor to 0  produces a non-real solution

So, the only real number solution is a = -1