  This paper deals with generators $\mathsf{A}$ of strongly continuous right
linear semigroups in Banach two-sided spaces whose set of scalars is an
arbitrary Clifford algebra $\mathit{C}\ell(0,n)$. We study the invertibility of
operators of the form $P(\mathsf{A})$, where $P(x)\in\mathbb{R}[x]$ is any real
polynomial, and we give an integral representation for $P(\mathsf{A})^{-1}$ by
means of a Laplace-type transform of the semigroup $\mathsf{T}(t)$ generated by
$\mathsf{A}$. In particular, we deduce a new integral representation for the
operator $(\mathsf{A}^2 - 2\mathrm{Re}(q) \,\mathsf{A} + |q|^2)^{-1}$. As an
immediate consequence, we also obtain a new proof of the well-known integral
representation for the $S$-resolvent operator of $\mathsf{A}$ (also called
spherical resolvent operator of $\mathsf{A}$).
