1. Factorize: $a^{5}+a^{4}+1$.
1. Let $\omega$ be any cube root of unity other than 1. Substituting $a=\omega$ into the polynomial, we have $\omega^{5}+\omega^{4}+1=\omega^{2}+\omega+1=0$. Therefore, we can conclude that the original polynomial contains the factor $a^{2}+a+1$. Using polynomial division, we find the other factor to be $a^{3}-a+1$, or we can use the method of splitting terms to find it, i.e., $a^{5}+a^{4}+1=a^{3}\left(a^{2}+a+1\right)-(a-1)\left(a^{2}+a+1\right)$.