Show that if $A$ is a retract of $X$ then for all $n \ge 0$ $H_n(X) \simeq H_n(A) \oplus H_n(X,A)$
So we have a retraction $r:X \to A$, which is surjective.
Consider the long exact sequence
$\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \to H_{n-1}(A) \cdots$
As $r$ is surjective we have that $H_n(X) \to H_n(X,A)$ is surjective, and hence $H_n(A) \to H_n(X)$ is injective. Thus we have a short exact sequence
$0 \to H_n(A) \to H_n(X) \to H_n(X,A) \to 0$
I am unsure how to go from the fact this is exact, to the result (assuming the above is correct!)