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!