I am taking a graduate class on Equation Theory and one of my homework questions asks me to "Determine all polynomials $p(x)$ such that $(x-1)(p(x+1))=(x+2)(p(x))$ and $p(2)=12$. A provided hint is to try carefully selected values of $x$.

I've been able to use Excel to "plug and chug" to find values for $x\in\mathbb{Z}\geq1$. ($p(x)=0$ for $x\in\mathbb{Z}<1$.) I used Excel to graph the values I found and I was given a trend line that approximated $p(x)=2x^3-2x$.

What might some approaches be to be able to find all such polynomials?
Thanks!