Let's analyze the work of both students to identify the mistake.
Student A:
- The original expression is \(-2(2x + 4y - 8)\).
- Distributing \(-2\):
- \(-2 \cdot 2x = -4x\)
- \(-2 \cdot 4y = -8y\)
- \(-2 \cdot -8 = +16\) (since multiplying a negative by a negative gives a positive)
Combining these results, we have: \[ -2(2x + 4y - 8) = -4x - 8y + 16 \] So Student A's result is correct.
Student B:
- The original expression is \(-(-8x + 7y - 10)\).
- Distributing the negative:
- \(-(-8x) = 8x\)
- \(-7y\) stays as \(-7y\)
- \(-(-10) = +10\)
Combining these results, we have: \[ -(-8x + 7y - 10) = 8x - 7y + 10 \] So Student B's result is incorrect.
The mistake is in the distribution of the negative sign, where \(7y\) was mistakenly kept negative.
Now, looking at Student B's work, the correct answer that should have been written is:
8x - 7y + 10 (derived from correcting B's distribution).
Based on the options provided:
- a: A (correct)
- b: B (incorrect)
- c: 8y (not applicable)
- d: +8x (not applicable)
Since Student A is correct and Student B is incorrect, the student that made the mistake is B.
The correction is to change B's result to:
8x - 7y + 10.
However, since none of the options provided directly state that, the best fit given the context and the way the question is framed would be (b) B, acknowledging the mistake in Student B's work.