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.

Learn how to factorize the expression 27y^3 + 64 using the sum of cubes formula. This problem explores polynomial factorization concepts, including the sum of cubes formula and step-by-step factorization techniques. Suitable for advanced high school students (Grades 10-12).

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