1. Given the equations: $$x^2 + 3x + c = 0$$ $$x^2 + 3x + c + 2 = 0$$ 2. Condition for two real and distinct roots for the first equation is that the discriminant $\Delta$ must be positive: $$\Delta = b^2 - 4ac > 0$$ Here, $a = 1$, $b = 3$, and $c = c$ (constant). 3. Calculate the discriminant of the first equation: $$\Delta_1 = 3^2 - 4 \times 1 \times c = 9 - 4c > 0$$ 4. Rearranging this inequality: $$9 - 4c > 0 \Rightarrow 9 > 4c \Rightarrow c < \frac{9}{4} = 2.25$$ So for the first equation to have two distinct real roots, $c < 2.25$. 5. For the second equation $x^2 + 3x + c + 2 = 0$ to have two complex and non-real roots, its discriminant must be negative: $$\Delta_2 = 3^2 - 4 \times 1 \times (c + 2) < 0$$ 6. Calculate the discriminant of the second equation: $$\Delta_2 = 9 - 4(c + 2) < 0$$ $$9 - 4c - 8 < 0 \Rightarrow 1 - 4c < 0 \Rightarrow 1 < 4c \Rightarrow c > \frac{1}{4} = 0.25$$ 7. Combining both inequalities: $$c < 2.25 \quad \text{and} \quad c > 0.25$$ 8. The values of $c$ must satisfy: $$0.25 < c < 2.25$$ 9. Checking the options given: (a) 2 or 3 — 2 fits but 3 does not because it should be less than 2.25 (b) 2 or 1 — both 1 and 2 fit within the interval (c) -2 or -3 — neither fit since $c$ must be positive and greater than 0.25 (d) -2 or -1 — neither fit 10. Thus, the correct answer is: **(b) 2 or 1**