1. **State the problem:** Solve the inequality $$4|4x + 4| + 3 > -7$$. 2. **Understand the absolute value and inequality:** The absolute value expression $$|4x + 4|$$ is always non-negative, so $$4|4x + 4|$$ is also non-negative or zero. 3. **Isolate the absolute value term:** Subtract 3 from both sides: $$4|4x + 4| + 3 - 3 > -7 - 3$$ $$4|4x + 4| > -10$$ 4. **Analyze the inequality:** Since $$4|4x + 4|$$ is always $$\geq 0$$, it is always greater than $$-10$$. 5. **Conclusion:** The inequality holds for all real values of $$x$$. **Final answer:** $$\boxed{\text{All real numbers } x \in \mathbb{R}}$$