To determine the number of real solutions to the quadratic equation \( 4x^2 + x + 3 = 0 \), we can use the discriminant, which is given by the formula:
\[ D = b^2 - 4ac \]
For the given quadratic equation, the coefficients are:
- \( a = 4 \)
- \( b = 1 \)
- \( c = 3 \)
Now, we can calculate the discriminant:
\[ D = 1^2 - 4 \cdot 4 \cdot 3 = 1 - 48 = -47 \]
Since the discriminant \( D \) is negative, the quadratic equation has no real solutions.
Therefore, the correct response is:
no real solutions