To factor the expression \( 64x^2 + 16x + 1 \), we can use the method of factoring a perfect square trinomial.
First, we can check if the expression is a perfect square trinomial. A perfect square trinomial can be expressed in the form \( (ax + b)^2 \).
- Identify the first term: \( 64x^2 \) can be written as \( (8x)^2 \).
- Identify the last term: \( 1 \) can be written as \( (1)^2 \).
- The middle term \( 16x \) can be expressed as \( 2 \times (8x) \times (1) \), which matches the form of a perfect square.
Thus, \( 64x^2 + 16x + 1 \) can be factored as:
\[ (8x + 1)^2 \]
So,
\[ 64x^2 + 16x + 1 = (8x + 1)(8x + 1) = (8x + 1)^2. \]
Final answer:
\[ 64x^2 + 16x + 1 = (8x + 1)^2. \]
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