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Let's solve the given factoring problems from your image.

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### Question 12
1. **Expression:**  
   \( 64x^3 + 125 \)  
   This is a **sum of cubes**. Recall the formula:  
   \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

   Here:
   - \( a = 4x \) (since \( (4x)^3 = 64x^3 \))
   - \( b = 5 \) (since \( 5^3 = 125 \))

   Now, applying the formula:
   \[ 64x^3 + 125 = (4x + 5)((4x)^2 - (4x)(5) + 5^2) \]
   \[ = (4x + 5)(16x^2 - 20x + 25) \]

   **Answer:**  
   \( (4x + 5)(16x^2 - 20x + 25) \)

2. **Expression:**  
   \( 27a^3 - 8b^3 \)  
   This is a **difference of cubes**. Recall the formula:  
   \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

   Here:
   - \( a = 3a \) (since \( (3a)^3 = 27a^3 \))
   - \( b = 2b \) (since \( (2b)^3 = 8b^3 \))

   Now, applying the formula:
   \[ 27a^3 - 8b^3 = (3a - 2b)((3a)^2 + (3a)(2b) + (2b)^2) \]
   \[ = (3a - 2b)(9a^2 + 6ab + 4b^2) \]

   **Answer:**  
   \( (3a - 2b)(9a^2 + 6ab + 4b^2) \)

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### Question 13
1. **Expression:**  
   \( 27x^3 + 1 \)  
   This is a **sum of cubes**. Using the same formula as above:  
   \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

   Here:
   - \( a = 3x \) (since \( (3x)^3 = 27x^3 \))
   - \( b = 1 \) (since \( 1^3 = 1 \))

   Now, applying the formula:
   \[ 27x^3 + 1 = (3x + 1)((3x)^2 - (3x)(1) + 1^2) \]
   \[ = (3x + 1)(9x^2 - 3x + 1) \]

   **Answer:**  
   \( (3x + 1)(9x^2 - 3x + 1) \)

2. **Expression:**  
   \( 64a^3 - 125b^3 \)  
   This is a **difference of cubes**. Using the formula again:  
   \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

   Here:
   - \( a = 4a \) (since \( (4a)^3 = 64a^3 \))
   - \( b = 5b \) (since \( (5b)^3 = 125b^3 \))

   Now, applying the formula:
   \[ 64a^3 - 125b^3 = (4a - 5b)((4a)^2 + (4a)(5b) + (5b)^2) \]
   \[ = (4a - 5b)(16a^2 + 20ab + 25b^2) \]

   **Answer:**  
   \( (4a - 5b)(16a^2 + 20ab + 25b^2) \)

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Do you need further clarifications or step-by-step explanations for any part?

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Here are five related questions for practice:

1. How do you factor \( x^3 + 8 \) using the sum of cubes formula?
2. What is the factorization of \( 8x^3 - 27 \)?
3. Can the difference of squares formula be used on cubic expressions?
4. How do you factor the expression \( 8a^3 + 125b^3 \)?
5. What happens if the exponents in the expression are not perfect cubes?

**Tip:** Always double-check if an expression fits a known factoring pattern (like sum or difference of cubes) to make solving easier.

Factor the expressions: 64x^3 + 125, 27a^3 - 8b^3, 27x^3 + 1, and 64a^3 - 125b^3.

Learn how to factor cubic expressions such as 64x^3 + 125 and 27a^3 - 8b^3 using the sum and difference of cubes formulas. This guide is perfect for high school students studying advanced algebra and factoring techniques.

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