To determine the number of real solutions for the quadratic equation \(4x^2 + x + 3 = 0\), we can use the discriminant. The discriminant (\(D\)) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:
\[ D = b^2 - 4ac \]
In this equation, the coefficients are \(a = 4\), \(b = 1\), and \(c = 3\). Plugging in these values, we find the discriminant:
\[ D = (1)^2 - 4(4)(3) = 1 - 48 = -47 \]
Since the discriminant is negative (\(D < 0\)), the quadratic equation has no real solutions.
Therefore, the correct response is:
no real solutions