\( \cdot \operatorname { det } \left( \begin{array} { c c c } { 3 } & { - x } & { 5 } \\ { 1 } & { 3 } & { - x } \\ { 2 } & { 1 } & { 2 x } \end{array} \right) = 0 \)

To find the values of \( x \) for which the determinant of the given matrix equals zero, you can calculate the determinant and set it equal to zero. For the matrix \[ \begin{pmatrix} 3 & -x & 5 \\ 1 & 3 & -x \\ 2 & 1 & 2x \end{pmatrix}, \] the determinant can be calculated using the cofactor expansion or the rule of Sarrus for 3x3 matrices. This will lead to a polynomial expression in \( x \), and you can then solve the resulting equation for the values of \( x \). Once you find the determinant as a polynomial, you can factor or use the quadratic formula to identify the roots that satisfy the equation. These roots will give the specific values of \( x \) for which the determinant is zero.