<s>[INST] 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.
 [/INST] As I commented, you can exploit the symmetry.
$$\eqalign{
  & f(1 - x) + {x^2}f(x) = 2x - {x^4}  \cr 
  & f(x) + {\left( {1 - x} \right)^2}f(1 - x) = 2\left( {1 - x} \right) - {\left( {1 - x} \right)^4} \cr} $$
Now you get
$$\eqalign{
  & f(1 - x) + {x^2}f(x) = 2x - {x^4}  \cr 
  & f(1 - x) = \frac{{2\left( {1 - x} \right) - {{\left( {1 - x} \right)}^4} - f\left( x \right)}}{{{{\left( {1 - x} \right)}^2}}} \cr} $$
So all you have to do is solve for $f(x)$ in 
$$2x - {x^4} - {x^2}f(x) = \frac{{2\left( {1 - x} \right) - {{\left( {1 - x} \right)}^4} - f\left( x \right)}}{{{{\left( {1 - x} \right)}^2}}}$$
</s>