To verify the equation [tex]\( x^2 + x - 12 = (x - 3)(x + 4) \)[/tex], we need to check if the factored form expands to the original polynomial correctly. The factored form can be expanded as follows:
[tex]\[
(x - 3)(x + 4) = x(x + 4) - 3(x + 4) = x^2 + 4x - 3x - 12 = x^2 + x - 12
\][/tex]
This shows that [tex]\( x^2 + x - 12 \)[/tex] indeed equals [tex]\( (x - 3)(x + 4) \)[/tex].
Now we will analyze each student's verification:
1. Damien:
- Damien checks if the constant term obtained by multiplying the factors [tex]\((x - 3)\)[/tex] and [tex]\((x + 4)\)[/tex] is correct.
- Damien states [tex]\(3(-4) = -12\)[/tex].
- This is correct since [tex]\((x - 3)(x + 4)\)[/tex] should give [tex]\(-12\)[/tex] when calculated as the product of [tex]\(-3\)[/tex] and [tex]\(4\)[/tex].
2. Lauryn:
- Lauryn checks if the sum of the coefficients of [tex]\(x\)[/tex] from the factors is correct.
- Lauryn states [tex]\(-4 + 3 = -1\)[/tex].
- This is also correct since combining the terms [tex]\(-4x\)[/tex] from [tex]\(x + 4\)[/tex] and [tex]\(3x\)[/tex] from [tex]\(x - 3\)[/tex] gives [tex]\(x\ (=\ -4 + 3)\)[/tex].
3. Rico:
- Rico checks both the product and sum of the factors.
- Rico states [tex]\(-4(3) = -12\)[/tex] and [tex]\(-4 + 3 = -1\)[/tex].
- Both conditions are correct as explained previously: the product gives [tex]\(-12\)[/tex] and the sum of the coefficients gives [tex]\(x\)[/tex].
4. Latisha:
- Latisha checks both the product and sum of the factors, but uses [tex]\(4(-3) = -12\)[/tex] and [tex]\(-3 + 4 = 1\)[/tex].
- Although the product gives [tex]\(-12\)[/tex], the sum part of her verification is incorrect. [tex]\(-3 + 4\)[/tex] should yield [tex]\(1\)[/tex], but the correct expected sum is [tex]\(-1\)[/tex].
Based on these verifications, we conclude:
- Damien, Lauryn, and Rico have correctly verified that [tex]\( x^2 + x - 12 = (x - 3)(x + 4) \)[/tex].
- Latisha, however, made an error in her summation step.
Thus, the correctness standings are:
- Damien: Correct
- Lauryn: Correct
- Rico: Correct
- Latisha: Incorrect
