How many real solutions does the following quadratic equation have?

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**.