1. **State the problem:** Solve the inequality $$x^2 + 5x + 6 < 0$$ using roots and test points. 2. **Find the roots:** To find the roots, solve the quadratic equation $$x^2 + 5x + 6 = 0$$. 3. **Factor the quadratic:** $$x^2 + 5x + 6 = (x + 2)(x + 3)$$ 4. **Set each factor to zero:** $$x + 2 = 0 \Rightarrow x = -2$$ $$x + 3 = 0 \Rightarrow x = -3$$ 5. **Roots are $$x = -3$$ and $$x = -2$$. These roots divide the number line into three intervals: $$(-\infty, -3)$$, $$(-3, -2)$$, and $$(-2, \infty)$$. 6. **Test points:** Choose a test point from each interval to determine where the inequality holds. - For $$(-\infty, -3)$$, choose $$x = -4$$. - For $$(-3, -2)$$, choose $$x = -2.5$$. - For $$(-2, \infty)$$, choose $$x = 0$$. 7. **Evaluate the inequality at each test point:** - At $$x = -4$$: $$(-4)^2 + 5(-4) + 6 = 16 - 20 + 6 = 2 > 0$$ (False for $$< 0$$) - At $$x = -2.5$$: $$(-2.5)^2 + 5(-2.5) + 6 = 6.25 - 12.5 + 6 = -0.25 < 0$$ (True) - At $$x = 0$$: $$0 + 0 + 6 = 6 > 0$$ (False) 8. **Conclusion:** The inequality $$x^2 + 5x + 6 < 0$$ holds true only in the interval $$(-3, -2)$$. **Final answer:** $$\boxed{(-3, -2)}$$