The homework question is:
If $I,J$ are ideals of $R$, let $IJ$ be the set of all sums of elements of the form $ij$, where $i \in I$, $j \in J$. Prove that $IJ$ is an ideal of $R$.
The phrase "the set of all sums of elements of the form $ij$" is confusing me. Does he mean strictly that $i_1j_1 + i_2j_2 \in IJ$, and $i + j \notin IJ$?
No, he means that $IJ$ is the smallest ideal such that all products $xy (x\in I, y\in J)$ are contained in it, and this is is the set $$ \{x_1y_1+x_2y_2+\dots+x_ky_k | k\in \mathbb{N}, x_\nu \in I, y_\mu \in J\} $$ You must show that the sum of two such elements is again in this set, and a multiple of such a sum is again such a sum.
If $x\in I$ and $y\in J$ then no claim is made about $x+y$.