\( x _ { 1 } x _ { 2 } = - 9 \pm \frac { \sqrt { 9 ^ { 2 } - 4 \cdot ( - 3 ) \cdot 12 } } { 2 \cdot - 3 } \)

To solve the expression \( x_1 x_2 = -9 \pm \frac{\sqrt{9^2 - 4 \cdot (-3) \cdot 12}}{2 \cdot -3} \), we will first simplify the expression inside the square root and then calculate the entire expression. 1. **Calculate the discriminant**: \[ 9^2 - 4 \cdot (-3) \cdot 12 = 81 + 144 = 225 \] 2. **Calculate the square root**: \[ \sqrt{225} = 15 \] 3. **Substitute back into the expression**: \[ x_1 x_2 = -9 \pm \frac{15}{2 \cdot -3} \] 4. **Calculate the denominator**: \[ 2 \cdot -3 = -6 \] 5. **Now substitute this into the expression**: \[ x_1 x_2 = -9 \pm \frac{15}{-6} \] 6. **Simplify \(\frac{15}{-6}\)**: \[ \frac{15}{-6} = -\frac{5}{2} \] 7. **Now substitute this back into the expression**: \[ x_1 x_2 = -9 \pm \left(-\frac{5}{2}\right) \] 8. **This gives us two cases**: - Case 1: \[ x_1 x_2 = -9 - \frac{5}{2} = -\frac{18}{2} - \frac{5}{2} = -\frac{23}{2} \] - Case 2: \[ x_1 x_2 = -9 + \left(-\frac{5}{2}\right) = -9 - \frac{5}{2} = -\frac{18}{2} - \frac{5}{2} = -\frac{23}{2} \] In both cases, we find that: \[ x_1 x_2 = -\frac{23}{2} \] Thus, the final result is: \[ \boxed{-\frac{23}{2}} \]