= Solution
The <fundamental theorem of algebra> factors $q(x,1)$ into linear factors over $\mathbb C$; homogenizing supplies factors of $y$ if the degree is less than three. Thus every nonzero homogeneous cubic is $cL_1L_2L_3$ for linear forms $L_i$, with $c\ne0$ absorbed into one factor.
For pairwise nonproportional factors, part (c) assigns three distinct tangent lines $\ell_i=\mathbb P(L_i\cdot\operatorname{span}\{x,y\})$ to $C$. Their pairwise intersections are $[L_1L_2]$, $[L_1L_3]$, and $[L_2L_3]$. There cannot be a common point of all three: a nonzero quadratic cannot be divisible by three pairwise nonproportional linear factors, by <unique factorization>. Hence the three lines form a triangle circumscribed about $C$.
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