Answer
$(3y+1)(9y^2-3y+1)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
27y^3+1
,$ use the factoring of the sum/difference of $2$ cubes.
$\bf{\text{Solution Details:}}$
Using $(a\pm b)(a^2\mp ab+b^2)$ or the factoring of the sum/difference of $2$ cubes, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(3y+1)[(3y)^2-3y(1)+(1)^2]
\\\\=
(3y+1)(9y^2-3y+1)
.\end{array}