Answer
$(2c+d)(4c^2-2cd+d^2)$
Work Step by Step
The expressions $
8c^3
$ and $
d^3
$ are both perfect cubes (the cube root is exact). Hence, the given expression $
8c^3+d^3
$ is a $\text{
sum
}$ of $2$ cubes.
Using the factoring of the sum or difference of $2$ cubes which is given by
\begin{align*}
a^3+b^3&=(a+b)(a^2-ab+b^2)
\\&\text{ or }\\
a^3-b^3&=(a-b)(a^2+ab+b^2)
,\end{align*}the expression above is equivalent to
\begin{align*}
&
(2c)^3+(d)^3
\\&=
(2c+d)[(2c)^2-(2c)(d)+(d)^2]
\\&=
(2c+d)(4c^2-2cd+d^2)
.\end{align*}
Hence, the factored form of $
8c^3+d^3
$ is $
(2c+d)(4c^2-2cd+d^2)
$.
