Answer
$(p+1)(p^2-p+1)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
p^3+1
,$ use the factoring of the sum or difference of $2$ cubes.
$\bf{\text{Solution Details:}}$
The expressions $
p^3
$ and $
1
$ are both perfect cubes (the cube root is exact). Hence, $
p^3+1
$ is a $\text{
sum
}$ of $2$ cubes. Using the factoring of the sum or difference of $2$ cubes which is given by $a^3+b^3=(a+b)(a^2-ab+b^2)$ or by $a^3-b^3=(a-b)(a^2+ab+b^2)$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(p)^3+(1)^3
\\\\=
(p+1)[(p)^2-p(1)+(1)^2]
\\\\=
(p+1)(p^2-p+1)
.\end{array}