I don’t think you’ve even done part (a) properly.

Recall that the coefficient of $\displaystyle x^k$ in $\displaystyle fg$ is

$\displaystyle \sum_{i\,=\,0}^ka_ib_{k-i}$

When $\displaystyle k=m+n,$ the sum is just $\displaystyle a_mb_n$ because $\displaystyle a_i=0$ for all $\displaystyle i>m$ and $\displaystyle b_{m+n-i}=0$ for all $\displaystyle i<m.$ Hence the coefficient of $\displaystyle x^{m+n}$ in $\displaystyle fg$ is $\displaystyle a_mb_n.$ And since $\displaystyle a_m\ne0,\ b_n\ne0$ and $\displaystyle R$ is an integral domain, $\displaystyle a_mb_n\ne0$ and so $\displaystyle fg$ is nonzero.

Do you understand this so far?