Question

Question was provided via image upload.

Explanation:

Analyze the expression

We evaluate the square of the binomial:

$$ (-4x + 9)^2 $$

This is a binomial squared, which always results in a Perfect Square Trinomial of the form:

$$ (a + b)^2 = a^2 + 2ab + b^2 $$

It does not represent a Difference of Squares, which is of the form \(a^2 - b^2 = (a-b)(a+b)\).

Expand the binomial

We expand the expression using the binomial square formula:

$$ (-4x + 9)^2 = (-4x)^2 + 2(-4x)(9) + (9)^2 $$

$$ (-4x + 9)^2 = 16x^2 - 72x + 81 $$

Kylie's calculation in the image is:

$$ (-4x + 9)^2 = (-4x)^2 + (9)^2 = 16x^2 + 81 $$

This calculation is incorrect because she omitted the middle term \(-72x\).

Evaluate Kylie's explanation

Kylie made two errors:

1. She incorrectly identified the expression as a Difference of Squares instead of a Perfect Square Trinomial.
2. She calculated the product incorrectly by omitting the middle term.

Therefore, Kylie did not understand that this is a perfect square trinomial, and she did not determine the product correctly.

Answer:

• Kylie is correct.
• Kylie correctly understood that it is a difference of squares, but she did not determine the product correctly.
• Kylie did not understand that this is a perfect square trinomial, and she did not determine the product correctly. (Correct answer)
• Kylie determined the product correctly, but she did not understand that this is a perfect square trinomial.