The given expression is:

\[
P^3 + 64q^3
\]

This is a sum of cubes, and it can be factored using the sum of cubes formula:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

In this case:

- \( a = P \)
- \( b = 4q \) (since \( 64q^3 = (4q)^3 \))

Now applying the formula:

\[
P^3 + 64q^3 = (P + 4q)(P^2 - P(4q) + (4q)^2)
\]

Simplifying the second factor:

\[
P^2 - P(4q) + (4q)^2 = P^2 - 4Pq + 16q^2
\]

Thus, the factored form of the expression is:

\[
P^3 + 64q^3 = (P + 4q)(P^2 - 4Pq + 16q^2)
\]

### Final Answer:

\[
P^3 + 64q^3 = (P + 4q)(P^2 - 4Pq + 16q^2)
\]

Would you like further explanation on how the sum of cubes works or any other details?

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Here are five related questions:

1. How do you factor a difference of cubes?
2. What is the general formula for factoring any sum of cubes?
3. Can you apply the sum of cubes to \(8a^3 + 27b^3\)?
4. How would you solve an equation involving a sum of cubes?
5. What are the real and complex roots of a sum of cubes equation?

**Tip:** Always look out for cube expressions where both terms are perfect cubes; they often follow simple factoring rules!

Learn how to factor the sum of cubes expression P^3 + 64q^3 using the sum of cubes formula. This step-by-step solution simplifies the problem into (P + 4q)(P^2 - 4Pq + 16q^2). Suitable for students in Grades 9-12 studying algebra and polynomial factoring.

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