To determine which statement is true regarding Adi's use of algebra tiles for the product [tex]\((-2x-2)(2x-1)\)[/tex], let’s go through the multiplication step-by-step. The expansion of the product is:
[tex]\[
(-2x - 2)(2x - 1)
\][/tex]
We can use the distributive property (FOIL method) to expand this:
1. First Terms:
[tex]\[
(-2x) \cdot (2x) = -4x^2
\][/tex]
2. Outer Terms:
[tex]\[
(-2x) \cdot (-1) = 2x
\][/tex]
3. Inner Terms:
[tex]\[
(-2) \cdot (2x) = -4x
\][/tex]
4. Last Terms:
[tex]\[
(-2) \cdot (-1) = 2
\][/tex]
Now, combining these results:
[tex]\[
-4x^2 + 2x - 4x + 2
\][/tex]
Simplify by combining like terms:
[tex]\[
-4x^2 - 2x + 2
\][/tex]
Given this distribution and simplification process, let's consider the statements about Adi's use of algebra tiles:
1. She used the algebra tiles correctly.
- This statement would be true if there were no errors in her use of algebra tiles. However, given the problem suggests there might be errors, we should examine the other statements.
2. She did not represent the two original factors correctly on the headers.
- This suggests that there might be an error in how Adi set up the factors [tex]\((-2x-2)\)[/tex] and [tex]\((2x-1)\)[/tex] on the headers. If there was a misrepresentation here, it would lead to incorrect results.
3. The signs on some of the products are incorrect.
- Given that we have both positive and negative terms involved, it's possible Adi could have made an error with the signs during the multiplication process.
4. Some of the products do not show the correct powers of [tex]\(x\)[/tex].
- This would mean there were errors in identifying the correct powers of [tex]\(x\)[/tex] in the resulting product terms. However, powers of [tex]\(x\)[/tex] in this specific example are straightforward ([tex]\(x^2\)[/tex] and [tex]\(x\)[/tex]), and significant errors in powers are less common without apparent indication.
Based on these possibilities and considering common errors in working with algebra tiles, it is plausible that the most likely mistake involves sign errors, as they are frequent and can easily occur when managing multiple negative terms.
Thus, the most accurate conclusion regarding the errors involves signs:
The signs on some of the products are incorrect.
Thereby, the correct diagnosis of Adi's use of algebra tiles is that:
[tex]\[
\boxed{3}
\][/tex]
The signs on some of the products are incorrect.
