Final answer:
Option (d) (25x² - 16xy² + 9y²) is the correct answer, as it fits the pattern of a perfect square trinomial and can be factored into (5x - 3y)^2.
Step-by-step explanation:
A perfect square trinomial is a special form of quadratic expression that can be factored into a binomial squared. The general form of a perfect square trinomial is (ax)^2 ± 2abx + (by)^2. To identify the perfect square trinomial, the first and last terms must be perfect squares, and the middle term must be twice the product of the square roots of these terms with the same sign.
Looking at the options provided:
- (a) (25x² + 16xy² - 9y²) is not a perfect square trinomial because the middle term is not twice the product of the square roots of the first and last terms.
- (b) (25x² - 16xy² + 9y²) - This doesn't fit the definition since the middle term should be positive if both the first and last terms are positive squares.
- (c) (25x² + 30xy² - 9y²) is not a perfect square trinomial for the same reason as (a); the middle term is incorrect.
- (d) (25x² - 16xy² + 9y²) is the correct answer because it follows the pattern of a perfect square trinomial: (5x)^2 - 2(5x)(3y) + (3y)^2, which can be factored as (5x - 3y)^2.