How many real solutions does the quadratic equation $x^2 + x + 5 = 0$ have?

Compute the discriminant: $b^2 - 4ac = 1^2 - 4(1)(5) = 1 - 20 = -19$. The discriminant is negative, so the square root is not a real number and the equation has no real solution. A negative discriminant always rules out real roots.