Example 3. Factorize: $a^{5}+a+1$.

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Solve the equation $\begin{aligned} & \left(a^{5}-a^{2}\right)+\left(a^{2}+a+1\right) \\ & =\left(a^{2}+a+1\right)\left(a^{3}-a^{2}+1\right) .\end{aligned}$