1. **State the problem:** Solve the inequality $$x^2 + 7x + 12 \geq 0$$. 2. **Formula and rules:** To solve quadratic inequalities, first find the roots of the quadratic equation $$x^2 + 7x + 12 = 0$$ by factoring or using the quadratic formula. Then analyze the sign of the quadratic expression in the intervals determined by the roots. 3. **Find the roots:** Factor the quadratic: $$x^2 + 7x + 12 = (x + 3)(x + 4) = 0$$ So the roots are $$x = -3$$ and $$x = -4$$. 4. **Analyze intervals:** The roots divide the number line into three intervals: - $$(-\infty, -4)$$ - $$(-4, -3)$$ - $$(-3, \infty)$$ 5. **Test each interval:** - For $$x < -4$$, pick $$x = -5$$: $$(x + 3)(x + 4) = (-5 + 3)(-5 + 4) = (-2)(-1) = 2 \geq 0$$ (True) - For $$-4 < x < -3$$, pick $$x = -3.5$$: $$(x + 3)(x + 4) = (-3.5 + 3)(-3.5 + 4) = (-0.5)(0.5) = -0.25 < 0$$ (False) - For $$x > -3$$, pick $$x = 0$$: $$(0 + 3)(0 + 4) = 3 \times 4 = 12 \geq 0$$ (True) 6. **Include roots:** Since the inequality is $$\geq 0$$, include points where the expression equals zero, i.e., $$x = -4$$ and $$x = -3$$. 7. **Final solution:** $$x \in (-\infty, -4] \cup [-3, \infty)$$ This means the quadratic expression is non-negative for all $$x$$ less than or equal to $$-4$$ and for all $$x$$ greater than or equal to $$-3$$.