Solve this equation and select the choice that contains all of the solutions. \[ x^{2}+2 x+10=0 \]

**Step 1. Identify the coefficients.** The quadratic equation is \[ x^2 + 2x + 10 = 0. \] Here, \( a = 1 \), \( b = 2 \), and \( c = 10 \). **Step 2. Compute the discriminant.** The discriminant \( \Delta \) is given by \[ \Delta = b^2 - 4ac. \] Substitute the values: \[ \Delta = 2^2 - 4(1)(10) = 4 - 40 = -36. \] **Step 3. Apply the quadratic formula.** The quadratic formula is \[ x = \frac{-b \pm \sqrt{\Delta}}{2a}. \] Substitute the values: \[ x = \frac{-2 \pm \sqrt{-36}}{2}. \] **Step 4. Simplify the expression.** Since \(\sqrt{-36} = 6i\) (where \( i \) is the imaginary unit), the equation becomes: \[ x = \frac{-2 \pm 6i}{2}. \] Divide numerator and denominator by 2: \[ x = -1 \pm 3i. \] **Step 5. State the solutions.** The solutions to the equation are: \[ x = -1 + 3i \quad \text{and} \quad x = -1 - 3i. \]