  Let $\pi$ and $\tau$ be a smooth generic representation of ${\rm SO}_5$ and
${\rm GL}_2$ respectively over a non-archimedean local field. Assume that $\pi$
is irreducible and $\tau$ is irreducible or induced of Langlands' type. We show
that the $L$- and $\epsilon$-factors attached to $\pi\times\tau$ defined by the
Rankin-Selberg integrals and the associated Weil-Deligne representation
coincide. Similar compatibility results are also obtained for the local factors
defined by the Novodvorsky's local zeta integrals attached to generic
representations of ${\rm GSp}_4\times{\rm GL}_2$.
