1. **State the problem:** Simplify and analyze the function $$y = \ln \left((x^3 + 1)^7\right) \cdot \left(x (1 - x)^{10}\right).$$ 2. **Recall logarithm properties:** The logarithm of a power can be simplified using $$\ln(a^b) = b \ln(a).$$ 3. **Apply the logarithm rule:** $$y = 7 \ln(x^3 + 1) \cdot x (1 - x)^{10}.$$ 4. **Interpret the expression:** The function is a product of $$7 \ln(x^3 + 1)$$ and $$x (1 - x)^{10}$$. 5. **Domain considerations:** - For $$\ln(x^3 + 1)$$ to be defined, $$x^3 + 1 > 0 \Rightarrow x > -1.$$ - For $$x (1 - x)^{10}$$, no additional domain restrictions beyond real numbers. 6. **Final simplified form:** $$y = 7 x (1 - x)^{10} \ln(x^3 + 1), \quad x > -1.$$