The problem is to factorize the expression $8x^3 + y^3$. This is a sum of cubes.

1. Problem Description

The problem is to factorize the expression $8x^3 + y^3$. This is a sum of cubes.

2. Solution Steps

We are given the expression $8x^3 + y^3$. We can rewrite $8x^3$ as $(2x)^3$. So, we have $(2x)^3 + y^3$.

We use the sum of cubes factorization formula:

$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$.

In this case, $a = 2x$ and $b = y$. Substituting these values into the formula gives us:

$(2x)^3 + y^3 = (2x + y)((2x)^2 - (2x)(y) + y^2)$.

Simplifying the terms inside the second parenthesis, we get:

$(2x)^3 + y^3 = (2x + y)(4x^2 - 2xy + y^2)$.

3. Final Answer

The factored form of the expression $8x^3 + y^3$ is $(2x + y)(4x^2 - 2xy + y^2)$.