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To solve this, let’s analyze each polynomial to see if it can be factored as a perfect square binomial.

1. **Option (a) \( x^2 + 4x + 64 \):**
   - To factor as a perfect square, the middle term should be \( 2 \times x \times \text{some number} \) such that we get 4 as the middle term.
   - However, this doesn’t satisfy the conditions of a perfect square because 4 is not twice the square root of 64.
   - **Not a perfect square binomial.**

2. **Option (b) \( x^2 - 64 \):**
   - This is a difference of squares, which factors as \( (x - 8)(x + 8) \).
   - It is factorable but not as a perfect square binomial.
   - **Not a perfect square binomial.**

3. **Option (c) \( x^2 + 64 \):**
   - This expression is a sum of squares, which cannot be factored further over the real numbers.
   - **Not a perfect square binomial.**

4. **Option (d) \( x^2 + 16x + 64 \):**
   - This matches the form of a perfect square trinomial \( (x + 8)^2 \) because \( 16 = 2 \times x \times 8 \) and \( 64 = 8^2 \).
   - **This is a perfect square binomial, as it factors to \( (x + 8)^2 \).**

**Answer: The correct option is (d) \( x^2 + 16x + 64 \).**

Would you like a further breakdown of this process? 

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Here are some related questions:
1. What are the conditions for factoring a trinomial as a perfect square?
2. How do you identify a difference of squares?
3. Why can’t sums of squares be factored over real numbers?
4. Can a polynomial with an odd power be a perfect square?
5. What are some common factoring techniques for polynomials?

**Tip:** To determine if a trinomial is a perfect square, check if the first and last terms are squares and if the middle term is twice the product of their square roots.

Which of the following polynomials can be factored as a perfect square binomial?

a) x^2 + 4x + 64
b) x^2 - 64
c) x^2 + 64
d) x^2 + 16x + 64

Explore which polynomial among x^2 + 4x + 64, x^2 - 64, x^2 + 64, and x^2 + 16x + 64 can be factored as a perfect square binomial. This exercise covers factoring techniques, including perfect square trinomials and differences of squares, suitable for students in Grades 8-10.

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