### Factorize: \begin{aligned} p^3 + q^3 + p + q \end{aligned}

$( p + q ) ( p^2 - pq + q^2 + 1 )$
1. We know that $$(a^3 + b^3) = (a + b) (a^2 - ab + b^2)$$ Thus \begin{aligned} p^3 + q^3 + p + q = \space& ( p^3 + q^3 ) + ( p + q ) \\ = \space& ( p + q ) ( p^2 - pq + q^2) + ( p + q ) \\ = \space& ( p + q ) ( p^2 - pq + q^2 + 1 ) \end{aligned}
2. Hence, $p^3 + q^3 + p + q = ( p + q ) ( p^2 - pq + q^2 + 1 ).$