Recently I came across a functional equation which always has a polynomial with integer coefficients solution.

Let $$ L_n(x)=(2 x+1)^2f(x+1)-4x(x+n+1)f(x)-((2 n+1)!!)^2\prod_{i=1}^n(x+i). $$ Problem: For any positive integer $n$, there exists a polynomial $f(x) $with integer coefficients of order $n$ satisfying $$ L_n(x)=0. $$ How to prove this question? Or can we find the solution to the above equation?

For $n=1$, the solution is $5x+4$.

For $n=2$, the solution is $89x^2+233x+128$.

For $n=3$, the solution is $3429x^3+18675x^2+30222x+13824$.

Hints, references and proof are all welcome!