1. **Problem statement:** Given the function $$f(x) = \frac{1}{2}x^2 e^{x+1}$$, find the domain of $$f(x)$$. 2. **Formula and rules:** The domain of a function is the set of all real numbers $$x$$ for which the function is defined. Here, $$f(x)$$ is a product of a polynomial $$\frac{1}{2}x^2$$ and an exponential function $$e^{x+1}$$. 3. **Intermediate work:** - The polynomial $$\frac{1}{2}x^2$$ is defined for all real $$x$$. - The exponential function $$e^{x+1}$$ is defined for all real $$x$$. 4. **Conclusion:** Since both parts are defined for all real numbers, the domain of $$f(x)$$ is all real numbers. **Final answer:** $$\text{Domain of } f(x) = (-\infty, +\infty)$$