= Solution
In the typical case, the line through $P,Q$, or their tangent when they coincide, is $v=mu+c$. Its intersections with $D$ have first coordinates given by
$$
u^3+(-2a-m^2)u^2+(b_1-2mc)u-c^2=0.
$$
Their product is $c^2$. The third intersection is $-(P+Q)$, which has the same first coordinate as $P+Q$. Consequently
$$
u(P)u(Q)u(P+Q)=c^2.
$$
Passing to the quotient by rational squares gives $q(P)q(Q)q(P+Q)=1$. Every <square class> is its own inverse, so
$$
\boxed{q(P+Q)=q(P)q(Q).}
$$
The excluded cases have their specified special values; the source asks only for this chord-and-tangent argument away from them.
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