Multiplying all terms by a^2 gives
a4+a3+a+1=0(a3+1)(a+1)=0(a+1)2(a2−a+1)=0a=−1 or a2−a+1=0
Note that the equation a2−a+1=0 has determinant Δ=(−1)2−4(1)(1)=−3<0, so a2−a+1=0 has no real solutions.
Hence, a = -1 is the only possibility. Then a5=(−1)5=−1.