\( c = \frac { - 4 \pm \sqrt { 4 ^ { 2 } - 4 ( 6 ) ( 4 ) } } { 2 ( 6 ) } \)

To solve the expression \[ c = \frac{-4 \pm \sqrt{4^2 - 4(6)(4)}}{2(6)}, \] we will first simplify the expression inside the square root. 1. Calculate \(4^2\): \[ 4^2 = 16. \] 2. Calculate \(4(6)(4)\): \[ 4(6)(4) = 96. \] 3. Now substitute these values back into the expression under the square root: \[ 4^2 - 4(6)(4) = 16 - 96 = -80. \] 4. Now substitute this back into the equation for \(c\): \[ c = \frac{-4 \pm \sqrt{-80}}{12}. \] 5. Simplify \(\sqrt{-80}\): \[ \sqrt{-80} = \sqrt{80} \cdot i = \sqrt{16 \cdot 5} \cdot i = 4\sqrt{5} \cdot i. \] 6. Substitute this back into the equation for \(c\): \[ c = \frac{-4 \pm 4\sqrt{5}i}{12}. \] 7. Simplify the fraction: \[ c = \frac{-4}{12} \pm \frac{4\sqrt{5}i}{12} = -\frac{1}{3} \pm \frac{\sqrt{5}}{3}i. \] Thus, the final result is: \[ c = -\frac{1}{3} \pm \frac{\sqrt{5}}{3}i. \]