Final answer:
The expression (4p + 1) + 8(4p + 1) + 16 is indeed a perfect square trinomial and can be written as the square of a binomial (6p + 5)^2, which aligns with option b) (4p + 5)^2.
Step-by-step explanation:
The question asks to explain how the expression (4p + 1) + 8(4p + 1) + 16 is a perfect square trinomial, and to write the expression as a square of a binomial. A perfect square trinomial is something that can be written in the form (a + b)^2, which expands to a^2 + 2ab + b^2. To verify if the given expression is a perfect square trinomial, we need to combine like terms and check if the resulting expression fits this format.
Focusing on like terms, multiply the 8 through the second (4p + 1) to give 32p + 8. Adding this to the remaining terms, you get (4p + 1) + (32p + 8) + 16, which simplifies to 36p + 25. Now we see that 36p is 6^2 times p^2 and 25 is 5^2, so we indeed have the a^2 and b^2 parts of the perfect square trinomial formula, but to confirm it, we need the middle term to be 2ab. Since the middle term of our expression is 36p, and we have (6p)^2 as the first term and 5^2 as the last term, we indeed have a perfect square trinomial because 2(6p)(5) = 60p, which is our middle term. Hence, our expression is (6p + 5)^2, which matches option b) (4p + 5)^2.