= Solution
Write a polynomial $f\in W$ by its coefficients. Evaluation of $f$ and of each first partial derivative at $p$ is polynomial in these coefficients and in the coordinates of $p$. Consequently the bad incidence locus
$$
B=\left\{(f,p):f(p)=1,
\quad f_{x_1}(p)=f_{x_2}(p)=f_{x_3}(p)=0\right\}
$$
is <Zariski-closed set>[Zariski closed], and $U=(W\times k^3)\setminus B$ is <Zariski-open set>[Zariski open]. Here a hypersurface is regarded as smooth at an ambient point outside it automatically.
In fact $B$ is empty. For homogeneous $f$ of degree $d>0$, the <Euler homogeneous function theorem> says
$$
p_1f_{x_1}(p)+p_2f_{x_2}(p)+p_3f_{x_3}(p)=df(p).
$$
If $(f,p)\in B$, its left side is zero while its right side is $d$, which is nonzero because $\operatorname{char}k=0$. Thus every nonzero level is covered by the <smoothness of a nonzero homogeneous level set>, and
$$
\boxed{U=W\times k^3,}
$$
which is certainly nonempty and Zariski open.
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