Proving coefficients of a polynomial form a subvariety
Your approach seems just fine to me. Consider the subvariety $X$ of $\mathbb{P}^2$ defined by a homogeneous equation $f(x,y,z)$ of degree $3$. Then $X$ is smooth if and only if, at every point $p$ of $X$, not all of the partial derivatives $\partial_{x_k} f(p)$ vanish. Notice that these partial derivatives can be expressed as a polynomial in the coefficients of the homogeneous equation that you started with, hence we're dealing with an open locus in the moduli space. The (reduced) complement, then, must be closed.