Solve the equation using the quadratic formula. \( 3 x^{2}=4 x+5 \) The solution set is \( \{\square \) (Simplify your answer. Type an exact answer, using radicals and \( i \) as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

To solve the equation \( 3x^2 - 4x - 5 = 0 \) using the quadratic formula, we identify \( a = 3 \), \( b = -4 \), and \( c = -5 \). The formula is given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] First, we calculate the discriminant: \[ b^2 - 4ac = (-4)^2 - 4 \cdot 3 \cdot (-5) = 16 + 60 = 76 \] Now we can substitute into the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{76}}{2 \cdot 3} = \frac{4 \pm \sqrt{76}}{6} \] Next, we simplify \( \sqrt{76} \): \[ \sqrt{76} = \sqrt{4 \cdot 19} = 2\sqrt{19} \] Thus, substituting back, we have: \[ x = \frac{4 \pm 2\sqrt{19}}{6} \] Breaking this down further gives: \[ x = \frac{2 \pm \sqrt{19}}{3} \] Hence, the solution set is \[ \left\{ \frac{2 + \sqrt{19}}{3}, \frac{2 - \sqrt{19}}{3} \right\} \]