= Solution
Using <Feynman slash notation>, the symmetric tensor $p_\mu p_\nu$ kills the antisymmetric <commutator> of the <gamma matrices>. The <Clifford algebra> gives
$$
\boxed{\not p\,\not p=\tfrac12p_\mu p_\nu\{\gamma^\mu,\gamma^\nu\}=p^2I.}
$$
Consequently $(\not p)^{2r}=(p^2)^rI$ and $(\not p)^{2r+1}=(p^2)^r\not p$. For a scalar-coefficient <polynomial> $F(z)=\sum_nc_nz^n$, split its even and odd powers:
$$
\boxed{F(\not p)=A(p^2)\not p+B(p^2)I,\quad A(z)=\sum_rc_{2r+1}z^r,\quad B(z)=\sum_rc_{2r}z^r.}
$$
This reduction assumes scalar coefficients; arbitrary noncommuting matrix coefficients would not give the stated scalar functions $A,B$.
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