which products result in a perfect square trinomial? select three options. (-x + 9)(-x - 9) (xy + x)(xy + x)…

Explanation:

Step1: Recall perfect - square trinomial formula

A perfect - square trinomial is of the form $(a + b)^2=a^{2}+2ab + b^{2}$ or $(a - b)^2=a^{2}-2ab + b^{2}$, which can be written as $(a\pm b)(a\pm b)$.

Step2: Analyze option 1

For $(-x + 9)(-x - 9)$, this is in the form of $(a + b)(a - b)=a^{2}-b^{2}$ (difference - of - squares formula), where $a=-x$ and $b = 9$, so it is not a perfect - square trinomial.

Step3: Analyze option 2

For $(xy + x)(xy + x)=(xy + x)^{2}$. Using the formula $(a + b)^2=a^{2}+2ab + b^{2}$ with $a = xy$ and $b=x$, we get $(xy)^{2}+2(xy)(x)+x^{2}=x^{2}y^{2}+2x^{2}y + x^{2}$, which is a perfect - square trinomial.

Step4: Analyze option 3

For $(2x-3)(-3 + 2x)=(2x - 3)^{2}$. Using the formula $(a - b)^2=a^{2}-2ab + b^{2}$ with $a = 2x$ and $b = 3$, we get $(2x)^{2}-2(2x)(3)+3^{2}=4x^{2}-12x + 9$, which is a perfect - square trinomial.

Step5: Analyze option 4

For $(16 - x^{2})(x^{2}-16)=-(x^{2}-16)(x^{2}-16)=-(x^{2}-16)^{2}$, but the negative sign in front makes it not in the standard form of a perfect - square trinomial (a perfect - square trinomial has a positive leading coefficient when written in standard form).

Step6: Analyze option 5

For $(4y^{2}+25)(25 + 4y^{2})=(4y^{2}+25)^{2}$. Using the formula $(a + b)^2=a^{2}+2ab + b^{2}$ with $a = 4y^{2}$ and $b = 25$, we get $(4y^{2})^{2}+2(4y^{2})(25)+25^{2}=16y^{4}+200y^{2}+625$, which is a perfect - square trinomial.

Answer:

B. $(xy + x)(xy + x)$ C. $(2x-3)(-3 + 2x)$ E. $(4y^{2}+25)(25 + 4y^{2})$