I don't know how to solve part (a) of exercise 1.5.8 in Robin Hartshorne's book Deformation Theory 1.5.8 (page 42):

Consider the Hilbert scheme of zero-dimensional closed subschemes of $\mathbb{P}^4_k$ of length $8$, the field $k$ is algebraically closed. There is one component of dimension $32$ that has a nonsingular open subset corresponding to sets of eight distinct points. We will exhibit another component containing a nonsingular open subset of dimension $25$.

(a) Let $R=k[x,y,z,w]$, let $\mathfrak{m}$ be a maximal ideal in this ring, and let $I=V+\mathfrak{m}^3$, where $V$ is a $7$-dimensional subvector space of $\mathfrak{m}^2/\mathfrak{m}^3$. Let $B=R/I$, and let $Z= \operatorname{Spec}(B)$ be the associated closed subscheme of $\mathbb{A}^4 \subset \mathbb{P}^4$. Show that the set of all such $Z$, as the point of its support ranges over $\mathbb{P}^4$, forms an irreducible $25$-dimensional subset of the Hilbert scheme $H=\operatorname{Hilb}^8(\mathbb{P}^4)$.

How can I show that this subscheme of the Hilbert scheme is irreducible? I have no idea even how to start. Is there a general strategy how to deal with that kind of questions to show that certain subscheme of a moduli space is irreducible? Clearly it suffice to construct an irreducible open dense subscheme sitting inside it, but I not see how to manage it in this exercise.