The determinant of a 2x2 matrix \(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\) is given by the formula: \(ad - bc\).

Given the matrix \(\begin{vmatrix} -x & 12 \\ -1 & 4 \end{vmatrix}\), we can calculate its determinant:

\(\text{Determinant} = (-x)(4) - (12)(-1) = -4x + 12\)

We are given that the determinant is -12:

\(-4x + 12 = -12\)

Subtract 12 from both sides:

\(-4x = -24\)

Divide by -4:

\(x = 6\)