Final answer:
The student's given factors (x-2)(x+1) do not multiply to produce the original trinomial 8x²-8x-16. The correct factors must maintain the coefficient of the leading term, which their answer does not do. Hence, the correct critique is that the factors do not multiply to give the original expression.
Step-by-step explanation:
When asked to factor the trinomial 8x²-8x-16, the provided answer of (x-2)(x+1) is incorrect. The correct factors must satisfy three conditions based on properties of multiplication: they must produce the original middle term when combined, they must multiply to give the constant term, and the coefficient of the squared term must be retained.
The student's answer fails to properly account for the coefficient of the squared term, which is 8 in the original trinomial. The correct factored form would maintain this coefficient and factor out a common factor if possible. Considering the constant term is -16 and the middle term is -8x, one correct factored form could be (4x+4)(2x-4), which simplifies to (4x+4)(2x-4) = 4(x+1)·2(x-2).
The fundamental mistake the student made is that the supplied factors do not multiply to give the original trinomial. Mathematically proving this is a matter of expansion. If we multiply (x-2)(x+1), we get x²-x-2, which does not equal the original trinomial 8x²-8x-16. Therefore, the correct answer is option 4) The factors do not multiply to give the original trinomial.