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To determine the number of real solutions for the quadratic equation \(4x^2 + x + 3 = 0\), we can use the discriminant, given by the formula:
\[
D = b^2 - 4ac
\]
For the equation \(ax^2 + bx + c = 0\), we have:
- \(a = 4\)
- \(b = 1\)
- \(c = 3\)
Now we can calculate the discriminant:
\[
D = (1)^2 - 4(4)(3) = 1 - 48 = -47
\]
Since the discriminant \(D\) is less than zero (\(D < 0\)), this indicates that the quadratic equation has **no real solutions**.
Thus, the answer is:
**no real solutions**.