Final answer:
For the quadratic equation 3x^2 + px + 4 = 0, two real distinct roots exist when p > 6.93 or p < -6.93. The possible integer values of p for which there are no real roots are between -6 and 6.
Step-by-step explanation:
The quadratic equation given is 3x^2 + px + 4 = 0, where p is a constant. To determine the nature of roots for a quadratic equation, we can use the discriminant, Δ = b^2 - 4ac. For the equation to have two real distinct roots, the discriminant must be greater than zero (Δ > 0).
The equation has a = 3, b = p, and c = 4. Plugging these values into the discriminant formula gives us Δ = p^2 - 4(3)(4) = p^2 - 48. To have two real distinct roots, we need p^2 - 48 > 0, which simplifies to p^2 > 48. Taking square roots of both sides, we find two sets of values of p: p > sqrt(48) or p < -sqrt(48), which approximate to p > 6.93 or p < -6.93.
For part (b), if the equation has no real roots, the discriminant must be less than zero (Δ < 0). This occurs when p^2 < 48. Since p is an integer, the possible integer values of p for which the equation has no real roots ranges from p = -6 to p = 6.