How the coefficient of $t^{m+q}$ in the product $g'h'$ is
$$g_0h_{m+q}+g_1h_{m+q-1}+ \cdots +g_mh_q+\cdots g_{m+q}h_0 ?$$
For example, if $g_2=x^2+1, h_3=x^3+x^2+1$, then what is the value of $g_0h_{m+q}$? Plz explain reason in general for other terms.
The source of the problem -
You have to decide the value of $m+q$. The given expression sums up the different ways to get the exponent of $t$ to be $m+q$. The first term comes from multiplying the constant term in $g$, which is $g_0$ by the $t^{m+q}$ term in $h$. The second comes from multiplying the linear term in $g$, which is $g_1t$ by the corresponding term in $h$, which is $h_{m+q-1}t^{m+q-1}$ and so on.
In your example, $g_0=1, g_1=0, g_2=1, h_0=1,h_2=1,h_3=1$