1. **State the problem:** Solve the inequality $$x^2 + 10x + 21 < 0$$. 2. **Formula and rules:** To solve a quadratic inequality, first find the roots of the quadratic equation $$x^2 + 10x + 21 = 0$$ by factoring or using the quadratic formula. The roots divide the number line into intervals. Test each interval to determine where the inequality holds. 3. **Factor the quadratic:** $$x^2 + 10x + 21 = (x + 3)(x + 7)$$ 4. **Find the roots:** Set each factor equal to zero: $$x + 3 = 0 \Rightarrow x = -3$$ $$x + 7 = 0 \Rightarrow x = -7$$ 5. **Determine intervals:** The roots split the number line into three intervals: - $$(-\infty, -7)$$ - $$(-7, -3)$$ - $$(-3, \infty)$$ 6. **Test each interval:** - For $$x < -7$$, pick $$x = -8$$: $$(x + 3)(x + 7) = (-8 + 3)(-8 + 7) = (-5)(-1) = 5 > 0$$ - For $$-7 < x < -3$$, pick $$x = -5$$: $$(x + 3)(x + 7) = (-5 + 3)(-5 + 7) = (-2)(2) = -4 < 0$$ - For $$x > -3$$, pick $$x = 0$$: $$(0 + 3)(0 + 7) = 3 \times 7 = 21 > 0$$ 7. **Conclusion:** The inequality $$x^2 + 10x + 21 < 0$$ holds true for $$x$$ in the open interval $$(-7, -3)$$. **Final answer:** $$\boxed{(-7, -3)}$$