How many real solutions does the following quadratic equation have?

To determine the number of real solutions for the quadratic equation \(4x^2 + x + 3 = 0\), we can use the discriminant. For a quadratic equation in the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is given by:

\[
D = b^2 - 4ac
\]

In this case, \(a = 4\), \(b = 1\), and \(c = 3\). We can substitute these values into the discriminant formula:

\[
D = 1^2 - 4(4)(3)
\]

Calculating this:

\[
D = 1 - 48 = -47
\]

Since the discriminant \(D\) is negative (\(-47 < 0\)), the quadratic equation has no real solutions.

Thus, the correct response is:

**no real solutions**.