1. **Stating the problem:** We are given the function $$y = 3\sqrt{x^4 + 4x^3 + 5}$$ and we want to understand or analyze it. 2. **Understanding the function:** The function involves a square root of a polynomial expression multiplied by 3. The square root function is defined only when the expression inside is non-negative. 3. **Domain considerations:** We need to ensure the radicand (expression inside the square root) is $$\geq 0$$: $$x^4 + 4x^3 + 5 \geq 0$$ 4. **Checking the radicand:** Since $$x^4$$ grows very fast and is always non-negative, and $$5$$ is positive, the expression $$x^4 + 4x^3 + 5$$ is always positive for all real $$x$$. Thus, the domain is all real numbers. 5. **Summary:** The function is defined for all real $$x$$ and is given by: $$y = 3\sqrt{x^4 + 4x^3 + 5}$$ 6. **No further simplification:** The expression inside the root does not factor nicely to simplify the root further. **Final answer:** $$y = 3\sqrt{x^4 + 4x^3 + 5}$$ is defined for all real $$x$$.