Algebra

Multiplying all terms by a^2 gives 

$a^4 + a^3 + a + 1 = 0\\ (a^3 + 1)(a + 1) = 0\\ (a + 1)^2(a^2 - a + 1) = 0\\ a = -1\text{ or }a^2 - a + 1 = 0$

 

Note that the equation $a^2 - a +1 = 0$ has determinant $\Delta = (-1)^2 - 4(1)(1) = -3 < 0$, so $a^2 - a +1 = 0$ has no real solutions.

 

Hence, a = -1 is the only possibility. Then $a^5 = (-1)^5 = -1$.