1. **State the problem:** We need to analyze the function $$f(x) = \ln{(x+1)^2} \cdot \sqrt{3x+1}$$ and understand its behavior. 2. **Rewrite the function:** Using logarithm properties, $$\ln{(x+1)^2} = 2\ln{(x+1)}$$, so $$f(x) = 2\ln{(x+1)} \cdot \sqrt{3x+1}$$. 3. **Domain considerations:** - The argument of the logarithm must be positive: $$x+1 > 0 \Rightarrow x > -1$$. - The expression under the square root must be non-negative: $$3x+1 \geq 0 \Rightarrow x \geq -\frac{1}{3}$$. Combining these, the domain is $$x > -1$$ and $$x \geq -\frac{1}{3}$$, so the domain is $$\left[-\frac{1}{3}, \infty\right)$$. 4. **Behavior at domain boundaries:** - At $$x = -\frac{1}{3}$$, $$\sqrt{3(-\frac{1}{3})+1} = \sqrt{0} = 0$$, and $$\ln{(-\frac{1}{3}+1)} = \ln{\frac{2}{3}} < 0$$, so $$f(-\frac{1}{3}) = 2 \cdot \ln{\frac{2}{3}} \cdot 0 = 0$$. 5. **Summary:** - The function is defined for $$x \geq -\frac{1}{3}$$. - It is the product of a logarithmic function and a square root function. - The function equals zero at $$x = -\frac{1}{3}$$. 6. **Desmos graph function:** $$y = 2\ln{(x+1)} \cdot \sqrt{3x+1}$$. This completes the analysis of the function as requested.