Solution (source code)

= Solution

In $H^+$ the unstarred $S$ occurs only as the initial symbol and has precisely the alternatives
$$
S\to\varepsilon,\qquad S\to S^*.
$$
The first produces only $\varepsilon$. After the second, the renamed-derivation argument from part (b) produces exactly $\mathcal L(G)$. Therefore
$$
\boxed{\mathcal L(H^+)=\mathcal L(G)\cup\{\varepsilon\}.}
$$