To address the problem at hand:
Let's first examine Kylie’s explanation:
[tex]\[
(-4x + 9)^2 = (-4x)^2 + 9^2 = 16x^2 + 81
\][/tex]
This is incorrect. The expression given is a binomial square, not simply the sum of squares. When squaring a binomial [tex]\((a + b)^2\)[/tex], the correct expansion formula is:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
To apply this to the expression [tex]\((-4x + 9)^2\)[/tex], let’s identify:
[tex]\[
a = -4x \quad \text{and} \quad b = 9
\][/tex]
Now, using the binomial expansion formula:
[tex]\[
(-4x + 9)^2 = (-4x)^2 + 2(-4x)(9) + 9^2
\][/tex]
Calculating each term:
[tex]\[
(-4x)^2 = 16x^2
\][/tex]
[tex]\[
2(-4x)(9) = -72x
\][/tex]
[tex]\[
9^2 = 81
\][/tex]
Combining these, the correct trinomial expansion is:
[tex]\[
16x^2 - 72x + 81
\][/tex]
Therefore, Kylie did not understand that squaring a binomial results in a perfect square trinomial, which includes the middle term [tex]\(2ab\)[/tex]. She incorrectly thought it was simply the sum of the squares of the individual terms.
Based on this analysis:
[tex]\[
The statement that best describes Kylie's explanation is:
\][/tex]
[tex]\[
\boxed{\text{Kylie did not understand that this is a perfect square trinomial, and she did not determine the product correctly.}}
\][/tex]
