The expression provided is $x^2 + 16x + 64$.

To factor it completely, we need to find two numbers that multiply to 64 and add to 16.

The numbers 8 and 8 satisfy these conditions because:

• $8 \times 8 = 64$
• $8 + 8 = 16$

Thus, the expression can be factored as: $x^2 + 16x + 64 = (x + 8)(x + 8) = (x + 8)^2$

So, the complete factorization of $x^2 + 16x + 64$ is $(x + 8)^2$.

Would you like any more details or have any questions? Here are some related questions to consider:

1. How can you verify the factorization of a quadratic expression?
2. What are the steps to factor a quadratic trinomial in general?
3. What is the significance of a perfect square trinomial?
4. How does factoring relate to solving quadratic equations?
5. How would the factorization change if the middle term were negative?

Tip: Always check if a quadratic trinomial can be factored as a perfect square by comparing the middle term to twice the square root of the constant term.