determine all polynomials $P(x)$ such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial.

• clearly we have to show $(x+1)P(x-1)-(x-1)P(x)=c$ for all values of $x$ ($c$ is a real constant)
• i have tried this: see if $r$ is a root of $P(x)$ then $P(r)=0$ and then $(r+1)P(r-1)=c$. Now we consider $P(x)=a(x-r)(x-c)(x-d)\cdots(x-l)$, so we have $$(r+1)P(r-1)=(c+1)P(c-1)=(d+1)P(d-1)=\dots\\=(l+1)P(l-1)=\pm 2rcd\dots l=c$$ [because see $P(0)=P(-1)=\dfrac{c}{2}$ ]
• then i am stuck!!!!!!! see $P(x)=v$ is a solution where $v$ is a constant.

As Vanchinathan said in the comments,

Take $P(x)= \sum\limits_{n=0}^r a_n x^n$ then plug in this into the equation

you get $\sum\limits_{n=0}^r a_n [ (x+1)(x-1)^n-(x-1)x^n]=c$

so $2 a_0 + a_1 (x-1) + \cdots+ a_r [ (x+1)(x-1)^r-(x-1)x^r]=c$ and

then you create linear equations with the coefficients using polynomial equality with the RHS. I hope this helps a bit.