To determine how many real solutions the quadratic equation \(4x^2 + x + 3 = 0\) has, we will use the discriminant. The discriminant \(D\) of a quadratic equation of the form \(ax^2 + bx + c = 0\) is given by:
\[ D = b^2 - 4ac \]
In this case, \(a = 4\), \(b = 1\), and \(c = 3\). Plugging in these values, we calculate the discriminant:
\[ D = 1^2 - 4(4)(3) = 1 - 48 = -47 \]
Since the discriminant \(D\) is negative (\(-47 < 0\)), this indicates that the quadratic equation has no real solutions.
Therefore, the answer is:
no real solutions.