100 Follower Problem! Algebra

Let $m \neq 0$ be an integer. Find the number of polynomials $p(x)$ with real coefficients such that, for real $x,$

$\Large{\left( { x }^{ 3 }-m{ x }^{ 2 }+1 \right) p\left( x+1 \right) +\left( { x }^{ 3 }+m{ x }^{ 2 }+1 \right) p\left( x-1 \right) =2\left( { x }^{ 3 }-m{ x }^{ 2 }+1 \right) p\left( x \right) }.$