= Solution
Suppose $G=A*B$ were a nontrivial <free product>. Its <Bass-Serre tree> action has trivial edge stabilizers and no global fixed vertex. Part c makes $a$ elliptic. Since $a$ has infinite order, the fixed set of every nonzero power $a^r$ is a single vertex: it is nonempty, while fixing two vertices would fix the intervening edge and put the infinite-order element $a^r$ in a trivial edge stabilizer.
Let this vertex be $v$. The relation $ba^mb^{-1}=a^n$ gives
$$
b\operatorname{Fix}(a^m)=\operatorname{Fix}(a^n),
$$
and both sides are the singleton $\{v\}$. Thus $b$ also fixes $v$. Since $a$ and $b$ generate the <Baumslag-Solitar group>, the entire group fixes $v$, contradicting the Bass-Serre action of a nontrivial free product. Hence no such decomposition exists.
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