The given expression $27y^3 + 64$ is a sum of cubes. The sum of cubes can be factored using the formula:

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

Step-by-Step Factorization

1. Identify $a$ and $b$:

   [ a = \sqrt[3]{27y^3} = 3y, \quad b = \sqrt[3]{64} = 4 ]

2. Apply the sum of cubes formula:

   Substitute $a = 3y$ and $b = 4$ into the formula:

   $27y^3 + 64 = (3y + 4)\left((3y)^2 - (3y)(4) + 4^2\right)$

3. Simplify the expression:

   $(3y + 4)\left(9y^2 - 12y + 16\right)$

Final Factored Form

The factored form of $27y^3 + 64$ is:

$27y^3 + 64 = (3y + 4)(9y^2 - 12y + 16)$

This is the complete factorization of the given expression.

Would you like further details or have any questions about this process?

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Here are some related questions:

1. How do you factor a difference of cubes?
2. What is the significance of the sum of cubes in algebra?
3. How can you verify if a polynomial is a sum of cubes?
4. What are some practical examples where sum or difference of cubes is used?
5. How would you factor $64y^3 + 27$?

Tip: Remember that the middle term in the quadratic factor of a sum or difference of cubes will always be negative if the original expression is a sum of cubes.