How to prove $x^2f -(x+1)f' = x^2g -(x+1)g' \Rightarrow f = g$ for polynomials f,g with rational coefficients

Question

I'm trying to prove that

$x^2f -(x+1)f' = x^2g -(x+1)g' \Rightarrow f = g$

is true whenever f,g are two polynomials with rational coefficients

All I could do, however, is finding $f(-1) = g(-1)$, $f'(0) = g'(0)$, and $f''(0) = g''(0)$

I'm stuck here and not really sure how to advance

Also not sure how the "rational coefficients" is useful in this context

Answer 1

Hint: With the polynomial $h=f-g$, is is enough to show $x^2h=(x+1)h’ \implies h\equiv 0$. However this is evident from considering the degree of the LHS and RHS, as no finite degree polynomial will satisfy the condition.