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Using the discriminant formula b² - 4ac, the nature of roots of the the quadratic equation can be identified.

x² + 2x + 3 = 0

a = 1;  b = 2;  c = 3

Discriminant = (2)² - 4 (1)(3) 
                      = 4 - 12
                      = - 8

If the discriminant is < 0 or the discriminant is negative, there is no real solution because the solution is imaginary or complex number.

It also means that there is no x-intercept or the graph (parabola does not pass through x-axis.  Either the minimum or maximum of the graph is below or above the x axis.

ANSWER:  The nature of the roots of the given solution is NO REAL SOLUTIONS.