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


Answer:

^@ ( p + q ) ( p^2 - pq + q^2 + 1 ) ^@

Step by Step Explanation:
  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 ). ^@

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