Question

It is possible for a quadratic equation to have no real-number solutions:Solve : x2 + 5x + 3 = 0

Answer 1

Sure, that just means that the parabola never crosses the x axis but has a vertex above if it opens up or below if it opens down.

x^2 + 5 x + 3

x = [ -5 +/- sqrt (25 - 12) ] / 2

that has real roots because (b^2-4ac) is positive.
However try
x^2 + 2 x + 3 = 0

x = [ -2 +/- sqrt (4 - 12) ]/2

x = -1 +/- .5 sqrt(-8)
sqrt of -8 = i sqrt 8 = 2i sqrt(2)

x = -1 +/- i sqrt 2
the solutions are complex because they contain i, the sqrt of -1