100 Follower Problem! Algebra

Algebra Level 5

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

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

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