Let $w(x)$ be defined as such: $w(x) = \prod_{j=0}^{n}(x-x_j)$, where $x_j$'s are distinct real numbers, $j=0,1,...,n$.
Suppose there exists exactly one point $x_k$, $k \in {0,1,...,n}$ such that $x_k$ has multiplicity $\geq 2$. I mean that it's given that the other points have multiplicity $1$.
Can we rewrite the product as $w(x) = (x-x_k)^2\prod_{j=0, j\neq k}^{n}(x-x_j)$?
No, there is something wrong. If $x_k$ is of multiplicity at least two, this means there is some other index, say $x_\ell$ for which $x_k=x_\ell$. Which contradicts the assumption that $x_k$ is the only one of multiplicity 2.
If $x_k$ has multiplicity at leats 2, then the product can be written as $$w(x)=(x-x_k)(x-x_\ell)\prod_{j=0, j\neq k, j\neq \ell}^{n}(x-x_j),$$ or $$w(x)=(x-x_k)^2\prod_{j=0, j\neq k, j\neq \ell}^{n}(x-x_j),$$