For what value(s) of \( k \) will the equation \( -2 k=3 x^{2}+4 x \) have non-real roots? INSTRUCTION: Give your answer as an equation or inequality with a fraction (no decimal values). For example: \( k=4 \).

To determine the values of \( k \) for which the equation \( -2k = 3x^2 + 4x \) will have non-real roots, we need to establish the conditions under which the quadratic equation \( 3x^2 + 4x + 2k = 0 \) has non-real roots. Non-real roots occur when the discriminant \( D < 0 \). The discriminant for the quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ D = b^2 - 4ac \] In this case, \( a = 3 \), \( b = 4 \), and \( c = 2k \). Therefore, we calculate: \[ D = 4^2 - 4(3)(2k) = 16 - 24k \] Setting the discriminant less than zero for non-real roots, we have: \[ 16 - 24k < 0 \] Solving for \( k \): \[ 16 < 24k \] \[ \frac{16}{24} < k \] \[ \frac{2}{3} < k \] Thus, the value of \( k \) must be greater than \( \frac{2}{3} \). The solution can be stated as follows: \[ k > \frac{2}{3} \]