Given the factors of $2$ integers $a$ and $b$, can you find the factors of the sum of $a$ and $b$ without first multiplying the factors to get $a$ and $b$? This is just something I recently thought of and became curious about. If it is possible, how would one go about doing it?
Example: factors of $a: 3,7$ and factors of $b : 2,2,3 $. What are factors of $a + b$ without first doing $3 \cdot 7$ and $2 \cdot 2 \cdot 3$ and then adding them together and then factoring it?
EDIT: I slightly mis-worded my question and I think I fixed the wording.