= Solution
The defining <SU(3)> <representation> restricts to the given <matrix representation>. All its restricted <matrices> are real, so the <conjugate representation> restricts identically. Hence the first two <branchings> are
$$
\boxed{\mathbf3\downarrow_G=\chi_{-,+}\oplus E,\qquad
\overline{\mathbf3}\downarrow_G=\chi_{-,+}\oplus E.}
$$
For the <adjoint representation of SU(3)>, use $\mathbf3\otimes\overline{\mathbf3}=\mathbf1\oplus\mathbf8$. This follows by splitting endomorphisms into scalar and traceless <matrices>. Subtracting one trivial constituent from <2/c> gives
$$
\boxed{\mathbf8\downarrow_G=
\chi_{+,+}\oplus\chi_{+,-}\oplus\chi_{-,+}\oplus\chi_{-,-}\oplus2E.}
$$
As an independent check, its restricted <character> is $|\chi_R|^2-1=(8,0,0,0,0)$, the <regular character> of $G$. Thus this is precisely the <regular representation>, whose <representation multiplicities> equal the corresponding <dimensions>.
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