To solve the given problem $x^2 + 2x = i$, we need to find the complex solutions and determine the product of their real parts.

First, rewrite the equation as: $x^2 + 2x - i = 0$

To solve this quadratic equation, we use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 2$, and $c = -i$.

Substitute these values into the formula: $x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-i)}}{2 \cdot 1}$ $x = \frac{-2 \pm \sqrt{4 + 4i}}{2}$ $x = \frac{-2 \pm \sqrt{4(1 + i)}}{2}$ $x = \frac{-2 \pm 2\sqrt{1 + i}}{2}$ $x = -1 \pm \sqrt{1 + i}$

Now, let's find the square root of $1 + i$. To do this, we express $1 + i$ in polar form: $1 + i = \sqrt{2} \left( \cos\frac{\pi}{4} + i \sin\frac{\pi}{4} \right)$

Taking the square root: $\sqrt{1 + i} = \sqrt{\sqrt{2} \left( \cos\frac{\pi}{4} + i \sin\frac{\pi}{4} \right)}$ $\sqrt{1 + i} = \sqrt[4]{2} \left( \cos\frac{\pi}{8} + i \sin\frac{\pi}{8} \right)$

In rectangular form, this can be approximated as: $\sqrt{1 + i} \approx 1.0987 + 0.4551i$

Therefore, the solutions are: $x_1 = -1 + (1.0987 + 0.4551i) \approx 0.0987 + 0.4551i$ $x_2 = -1 - (1.0987 + 0.4551i) \approx -2.0987 - 0.4551i$

The real parts of the solutions are approximately $0.0987$ and $-2.0987$.

Finally, the product of their real parts is: $0.0987 \times -2.0987 \approx -0.2071$

Thus, the product of the real parts of the solutions is approximately $-0.2071$.

Would you like more details or have any questions?

Here are some questions you might want to explore next:

1. How do you solve a quadratic equation with complex coefficients?
2. Can you explain the polar form of a complex number?
3. What is the significance of the roots of a quadratic equation in complex analysis?
4. How can we verify the solutions of a quadratic equation?
5. What are some applications of complex numbers in engineering and physics?

Tip: When dealing with complex roots, converting complex numbers to their polar form often simplifies the calculation of roots and powers.