1. **State the problem:** Solve the equation $$x + \sqrt{11x + 44} = -4$$ for $x$. 2. **Isolate the square root term:** Move $x$ to the right side: $$\sqrt{11x + 44} = -4 - x$$ 3. **Important rule:** The expression inside the square root must be non-negative, and the right side must also be non-negative because a square root cannot equal a negative number. So, $$11x + 44 \geq 0 \quad \text{and} \quad -4 - x \geq 0$$ 4. **Solve inequalities:** $$11x + 44 \geq 0 \Rightarrow x \geq -4$$ $$-4 - x \geq 0 \Rightarrow x \leq -4$$ 5. **Combine inequalities:** The only possible $x$ is $x = -4$. 6. **Check $x = -4$ in the original equation:** $$-4 + \sqrt{11(-4) + 44} = -4 + \sqrt{-44 + 44} = -4 + 0 = -4$$ This matches the right side. 7. **Conclusion:** The solution is $$x = -4$$