Find the imaginary solutions to the following equation: \[ y=x^{2}+2 x+2 \]

Step 1 ：We are given the quadratic equation \(y=x^{2}+2 x+2\).

Step 2 ：We need to find the imaginary solutions to this equation.

Step 3 ：The solutions to a quadratic equation are given by the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where a, b, and c are the coefficients of the quadratic equation.

Step 4 ：In this case, a = 1, b = 2, and c = 2.

Step 5 ：The term under the square root, b^2 - 4ac, is called the discriminant. If the discriminant is negative, the solutions are imaginary.

Step 6 ：Substituting the values of a, b, and c into the discriminant, we get D = -4.

Step 7 ：Since the discriminant is negative, the solutions to the equation are complex numbers.

Step 8 ：Substituting the values of a, b, and c into the quadratic formula, we get x1 = (-1-1j) and x2 = (-1+1j).

Step 9 ：The solutions to the equation are x1 = -1 - i and x2 = -1 + i.

Step 10 ：Final Answer: The imaginary solutions to the equation are \(\boxed{-1 - i}\) and \(\boxed{-1 + i}\).