Solve for all solutions of \( x \) : \( 2 x^{2}+16 x+20=0 \) \( x=4 \pm \sqrt{6} \) \( x=-4 \pm \sqrt{26} \) \( x=-2 \pm \sqrt{6} \) \( x=-4 \pm \sqrt{6} \)

To solve the quadratic equation \( 2x^2 + 16x + 20 = 0 \), we can first simplify it by dividing all terms by 2, yielding: \[ x^2 + 8x + 10 = 0. \] Next, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 8 \), and \( c = 10 \). Calculating the discriminant: \[ b^2 - 4ac = 8^2 - 4 \cdot 1 \cdot 10 = 64 - 40 = 24. \] Now, applying the quadratic formula: \[ x = \frac{-8 \pm \sqrt{24}}{2 \cdot 1} = \frac{-8 \pm 2\sqrt{6}}{2} = -4 \pm \sqrt{6}. \] Thus, the solutions are: \[ x = -4 + \sqrt{6} \quad \text{and} \quad x = -4 - \sqrt{6}. \] So, the correct answers are \( x = -4 \pm \sqrt{6} \). When dealing with quadratic equations, it’s helpful to remember that the solutions can be expressed in various forms. One common mistake is miscalculation of the discriminant, which can drastically change the final answer. Always double-check arithmetic operations, especially with negative signs!