Final answer:
To determine which of the given trinomials is a perfect square, we need to check if the first and last terms of the trinomial are perfect squares and if the middle term is twice the product of the square roots of the first and last terms.
Step-by-step explanation:
To determine which of the given trinomials is a perfect square, we need to check if the first and last terms of the trinomial are perfect squares, and if the middle term is twice the product of the square roots of the first and last terms. Let's check each option:
a) x^2 + 6x + 9 - The first and last terms are perfect squares (x^2 and 9), and the middle term (6x) is twice the product of the square roots (2 * sqrt(x^2) * sqrt(9) = 6x). Therefore, option a) is a perfect square trinomial.
b) 3x^2 - 5x + 7 - The first term is not a perfect square, so option b) is not a perfect square trinomial.
c) 2x^2 + 4x + 2 - The first term is not a perfect square, so option c) is not a perfect square trinomial.
d) 4x^2 - 3x - 1 - The first term is not a perfect square, so option d) is not a perfect square trinomial.
Therefore, the answer is option a) x^2 + 6x + 9.