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 (\(D\)) from the quadratic formula.

The discriminant is given by the formula:

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

where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\).

For our equation:

- \(a = 4\)
- \(b = 1\)
- \(c = 3\)

Now, we can calculate the discriminant:

\[
D = (1)^2 - 4(4)(3)
\]
\[
D = 1 - 48
\]
\[
D = -47
\]

Since the discriminant \(D\) is less than 0, it indicates that there are no real solutions to the equation \(4x^2 + x + 3 = 0\).

Thus, the answer is:

**no real solutions**.