1. The problem asks to find the solution set of the inequality $$x^{2} - 5x - 14 < 0$$ where $$x \in \mathbb{R}$$. 2. To solve quadratic inequalities, we first find the roots of the corresponding quadratic equation $$x^{2} - 5x - 14 = 0$$. 3. Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$ where $$a=1$$, $$b=-5$$, and $$c=-14$$. 4. Calculate the discriminant: $$\Delta = (-5)^{2} - 4 \times 1 \times (-14) = 25 + 56 = 81$$. 5. Find the roots: $$x = \frac{5 \pm \sqrt{81}}{2} = \frac{5 \pm 9}{2}$$. 6. So the roots are: $$x_1 = \frac{5 - 9}{2} = \frac{-4}{2} = -2$$ $$x_2 = \frac{5 + 9}{2} = \frac{14}{2} = 7$$. 7. Since the parabola opens upwards (coefficient of $$x^2$$ is positive), the quadratic expression is less than zero between the roots. 8. Therefore, the solution set is all $$x$$ such that $$-2 < x < 7$$. 9. This corresponds to option d: $$\{x | -2 < x < 7\}$$.