Concerning Serre’s Intersection multiplicity

Question

I am trying to understand a statement in a proof.
The setup is $(R,m)$ is a $3$-dimensional regular local ring with infinite residue field, $\mathfrak{p}$ is a height-$2$ prime ideal and $x$ is not in $\mathfrak{p}$. Also, $\mathfrak{p}^{(m)}$ represents the $m$th symbolic power of $\mathfrak{p}.$ My difficulty is with the following equation: $$\chi(R/\mathfrak{p}^{(m)},R/(x))=\chi(R/(\mathfrak{p},x))\ell(R_{\mathfrak{p}}/\mathfrak{p}_{\mathfrak{p}}^m)$$ Is the $\chi$ on the right just the Hilbert-Samuel multiplicity of the quotient? And I am also not clear on how, $\ell$, the analytic spread comes in?