What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution.

Given, $x^2f(x)+f(1−x)=2x−x^4$
$f(-1) = 0$
$f(0) = 1$
$f(1) = 0$
$f(2) = -3$
$f(3) = - 8$
$f(4) = -15$
By careful observation one can see that the difference between the terms vanishes after the second stage
i.e. $f(x)$ is of the form $a{x^2} + b{x} +c$
i.e. $f(x) = a{x^2} + b{x} + c$ -------(1) 
i.e. $f(0) = c = 1$
i.e. $c = 1$ -------(2)
$f(1) = a + b + c = 0$
i.e. $f(1) = a + b + 1= 0$
i.e. $a + b = -1$ -------(3)
$f(2) = 4a + 2b + c = -3$
i.e. $4a + 2b + 1 = -3$
i.e. $4a + 2b = -2$
i.e. $2a + b = -2$ -------(4)
Solve equations $(3)$ and $(4)$ to get $a = -1$ and $b = 0$
i.e. $f(x) = {-1}{x^2} + {0}{x} + 1$
i.e. $f(x) = {-x^2} + 1$
