The equation \( x^2 + 1 = 0 \) has no real solutions because \( x^2 \geq 0 \) for all real numbers \( x \), meaning \( x^2 + 1 \) is always greater than 0. However, in the realm of complex numbers, the solutions are \( x = i \) and \( x = -i \), where \( i \) is the imaginary unit (i.e., \( i^2 = -1 \)).