If two groups have a solvable word, then their free product and direct product have a solvable word. I am having a rough time with this question. If there are any suggestions on how to start this, that would be very helpful.
To tell whether a word $w((g_1,h_1),\ldots,(g_n,h_n))$ in $G\times H$ is trivial, note that $$w\bigl((g_1,h_1),\ldots,(g_n,h_n)\bigr) = \bigl( w(g_1,\ldots,g_n),w(h_1,\ldots,h_n)\bigr),$$ and that $(x,y)\in G\times H$ is trivial if and only if $x$ is trivial and $y$ is trivial.
To tell whether a word in the free product $G*H$ is trivial, write it as a product $$g_1h_1g_2h_2\cdots g_mh_m$$ where $g_i\in G$, $h_i\in H$, $h_i\neq 1$ for $i=1,\ldots m-1$, and $g_i\neq 1$ for $i=2,\ldots,m$. This will be trivial if and only if $m=1$, $g_1=1$ and $h_1=1$.