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If the determinant of the matrix \[ \begin{vmatrix} x-4 & 2x & 2x\\ 2x & x-4 & 2x\\ 2x & 2x & x-4 \end{vmatrix} \] is written in the form \((A+Bx)(x-A)^2\), then what is the ordered pair \((A,B)\)?

1. (-4, 3)
2. (-4, 5)
3. (4, 5)
4. (-4, -5)

Correct answer: (-4, 3)

Solution

The determinant of the given matrix can be factored into the form (A + Bx)(x - A)^2, where the coefficients A and B are determined by evaluating the determinant and matching it to the specified form. In this case, the calculations reveal that A is -4 and B is 3, making the correct option (-4, 3).

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