1. **Stating the problem**: We have a binary operation $*$ on $\mathbb{R}$ defined by $a * b = a + b + 3$. 2. **Find the identity element $e$ if it exists:** The identity element satisfies $a * e = a$ for all $a \in \mathbb{R}$. Using the definition: $$a * e = a + e + 3 = a$$ Solving for $e$: $$a + e + 3 = a \implies e + 3 = 0 \implies e = -3$$ So the identity element is $e = -3$. 3. **Find the formula for the inverse of $a$ (denoted $a^{-1}$) if it exists:** The inverse $a^{-1} = x$ satisfies: $$a * x = e = -3$$ From the operation definition: $$a * x = a + x + 3 = -3$$ Solving for $x$: $$a + x + 3 = -3 \implies x = -3 - 3 - a = -6 - a$$ Therefore, the inverse of $a$ with respect to $*$ is: $$a^{-1} = -6 - a$$ Hence, the identity element is $-3$ and the inverse of $a$ is $-6 - a$.