Let $0 \to A \to B \to C \to 0$ be a short exact sequence of left $R$-modules. If $M$ is any left $R$-module, prove that there are exact sequences $$ 0 \to A \oplus M \to B \oplus M \to C \to 0 $$ and $$ 0 \to A \to B \oplus M \to C \oplus M \to 0. $$
(Original picture of the problem here.)
Proof Attempt: Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequence of left $R$-modules. Let $M$ be any left $R$-Module. We want to adjoin $M$ to the corresponding exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$.
At this point I'm noticing that $A \cong A \oplus M$ and $A \oplus M$ and $B \cong B \oplus M$ would need to true for the corresponding exact sequences $0 \rightarrow A \oplus M \rightarrow B \oplus M \rightarrow C \rightarrow0$, $0 \rightarrow A \rightarrow B \oplus M \rightarrow C \oplus M \rightarrow 0$.
However, my problem is trying to figure out how to implement the (internal) direct sums in the first exact sequence to get the two new exact sequences.
I would appreciate only advice and hints.