We can split up $SU(5)$ into real and imaginary parts as $U=U_R+iU_I$ and in doing so embed this in $SO(10)$ as $\begin{pmatrix} U_R & -U_I \\ U_I &U_R\end{pmatrix}$.
Hence we know that $SU(5)$ is a subgroup of $SO(10)$. We can also show that the embedding above is diagonalisable and hence hence branches to $5\oplus\bar{5}$.
The question then asks how the adjoint representation of $SO(10)$ branches under this embedding. So the dimension of the adjoint representation is the number of elements in the lie algebra so for $SO(10)$ the dimension is 45.
How do I work this out?