I have troubles 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 $8$. 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 $Z$ 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. Is there a general strategy how to deal with that kind of questions to show that certain subscheme of a moduli space is irreducible?