Solve for all solutions of x : \[ 9 x^{2}+54 x=18 \] \( x=-3 \pm i \sqrt{7} \) \( x=-3 \pm \sqrt{7} \) \( x=3 \pm \sqrt{11} \) \( x=-3 \pm \sqrt{11} \)

To solve for all solutions of the equation \(9x^{2} + 54x = 18\), we first rearrange it into standard quadratic form. Subtract 18 from both sides: \[ 9x^{2} + 54x - 18 = 0. \] Next, we can simplify this equation by dividing all terms by 9: \[ x^{2} + 6x - 2 = 0. \] Now we can apply the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=6\), and \(c=-2\). Plugging in these values: 1. Calculate the discriminant: \[ b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot (-2) = 36 + 8 = 44. \] 2. Substitute back into the quadratic formula: \[ x = \frac{-6 \pm \sqrt{44}}{2 \cdot 1} = \frac{-6 \pm 2\sqrt{11}}{2} = -3 \pm \sqrt{11}. \] So the solutions to the equation are: \[ x = -3 \pm \sqrt{11}. \] Now, comparing this with the options given, the correct solution is: \( x=-3 \pm \sqrt{11} \).