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The problem in your image is to factor the quadratic expression:

\[
16x^2 + 8x + 1
\]

### Step-by-step solution:

1. **Identify coefficients**:
   - \( a = 16 \) (coefficient of \( x^2 \)),
   - \( b = 8 \) (coefficient of \( x \)),
   - \( c = 1 \) (constant term).

2. **Check if it's a perfect square trinomial**:
   A perfect square trinomial takes the form \( (Ax + B)^2 \), where:
   \[
   (Ax + B)^2 = A^2x^2 + 2ABx + B^2
   \]
   Let's check if this matches the given expression.

   - \( A^2 = 16 \Rightarrow A = 4 \)
   - \( B^2 = 1 \Rightarrow B = 1 \)
   - Now, check if \( 2AB = 8 \):
     \[
     2(4)(1) = 8
     \]
     This condition is satisfied.

3. **Conclusion**:
   The given quadratic expression can be factored as a perfect square trinomial:
   \[
   16x^2 + 8x + 1 = (4x + 1)^2
   \]

Thus, the factorization is:

\[
(4x + 1)^2
\]

Would you like further details or explanations on any step? 

Here are some follow-up questions to explore the concept deeper:

1. Can you explain why the middle term must be \(2AB\) for a perfect square trinomial?
2. What is the general method to factor a trinomial that is not a perfect square?
3. How would you factor a quadratic equation with negative terms, like \(16x^2 - 8x + 1\)?
4. What role does the discriminant play in determining whether a quadratic expression can be factored easily?
5. Can you expand \( (4x + 1)^2 \) to verify the factorization?

**Tip**: Always check for common factors before factoring quadratics!

Learn how to factor the quadratic expression 16x^2 + 8x + 1 as a perfect square trinomial. This problem covers key algebraic concepts like factoring and identifying perfect square trinomials. Suitable for students in Grades 9-10.

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