Answer: (i) $A \to aA',\ A' \to S_1 \mid S_2$

Step 1 — left factoring

Given grammar:
$$A \to aS_1 \mid aS_2$$

The two alternatives share the common prefix a. We factor it out into a new nonterminal $A'$:

$$A \to aA'$$

$$A' \to S_1 \mid S_2$$

This is the standard left-factored (CFG) form that removes backtracking uncertainty, because after reading a the parser chooses between $S_1$ and $S_2$ by looking at the next input, not by backtracking.

Hence the correct answer is:

$$\boxed{\text{(i)}\ A \to aA',\ A' \to S_1 \mid S_2}$$