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The number of real solutions of a quadratic equation can be determined by calculating the discriminant. The discriminant (D) of a quadratic equation in the form ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
In the given equation, the coefficients are a = 4, b = 1, and c = 3. Calculating 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.