Solution
= Solution
Contract the <Kronecker deltas> in the first term to obtain $a_i b_i$. In the second term, $\delta_{ij}$ identifies the two $a$ components, while the two $b_k$ components already contract. Therefore
$$
\boxed{a_i b_k\delta_{km}\delta_{im}+a_i b_k a_j b_k\delta_{ij}
=\mathbf a\cdot\mathbf b+(\mathbf a\cdot\mathbf a)(\mathbf b\cdot\mathbf b)}.
$$