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:
\[ D = b^2 - 4ac \]
Here, \( a = 4 \), \( b = 1 \), and \( c = 3 \).
Now, calculate the discriminant:
\[ D = 1^2 - 4 \cdot 4 \cdot 3 = 1 - 48 = -47 \]
Since the discriminant \( D \) is negative (\( -47 < 0 \)), the quadratic equation has no real solutions.
Thus, the correct response is:
no real solutions.