The identity element for the given binary operation is zero, which means that for any real number \(a\), \(a * 0 = a\). We are looking for the inverse of 2 under this operation, denoted as \(2^{-1}\), such that \(2 * 2^{-1} = 0\).

Let's solve for \(2^{-1}\) using the given operation:

\(2 * 2^{-1} = 2 \cdot 2^{-1} + 2 + 2^{-1}\)

We want this expression to equal zero:

\(2 \cdot 2^{-1} + 2 + 2^{-1} = 0\)

Now, combine the terms with \(2^{-1}\):

\(2 \cdot 2^{-1} + 2^{-1} = -2\)

Simplify:

\(2^{-1} (2 + 1) = -2\)

\(2^{-1} = -2 / 3\)

So, the inverse of 2 under this operation is \(2^{-1} = -2/3\).