1. **Problem:** Given the binary operation $x * y = x + y + 2$, find the identity element $e$ such that for any $x$, $x * e = x$. 2. **Formula and rule:** The identity element $e$ satisfies the equation: $$x * e = x$$ Using the operation definition: $$x * e = x + e + 2$$ Set equal to $x$: $$x + e + 2 = x$$ 3. **Solve for $e$:** Subtract $x$ from both sides: $$e + 2 = 0$$ Then: $$e = -2$$ 4. **Explanation:** The identity element is the value that when combined with any $x$ under the operation $*$ returns $x$ itself. Here, $e = -2$ satisfies this condition. **Final answer:** The identity element is $-2$.