To determine how many real solutions the quadratic equation \(4x^2 + x + 3 = 0\) has, we can use the discriminant, which is calculated from the coefficients of the quadratic equation in the form \(ax^2 + bx + c = 0\). The discriminant is given by:
\[ D = b^2 - 4ac \]
For the equation \(4x^2 + x + 3 = 0\):
- \(a = 4\)
- \(b = 1\)
- \(c = 3\)
Now, we calculate the discriminant:
\[ D = 1^2 - 4 \cdot 4 \cdot 3 = 1 - 48 = -47 \]
Since the discriminant \(D\) is negative (\(-47 < 0\)), this means that the quadratic equation has no real solutions.
Therefore, the correct response is:
no real solutions