Is there a simpler function $f$ equivalent to $f(x)=af(x-1)^2+bf(x-1)+c$?

Question

I have been using the function $$f(x)=af(x-1)^2+bf(x-1)+c$$ for a project, but wanted to know if there was a closed form of the equation or a form of the function in relation to $f(0)$ or $f(1)$. If not, is there a way to solve for a, b, and c with 3 or 4 values of $f(x)$? I know that $$f(1)=af(0)^2+bf(0)+c$$ $$f(2)=a^3f(0)^4+2a^2bf(0)^3+2a^2cf(0)^2+ab^2f(0)^2+2abcf(0)+ac^2+abf(0)^2+b^2f(0)+bc+c$$ and so on, but it would be much easier if there was a simpler way of evaluating this equation.