Find $f(x)$ satisfying the functional equation $x^2{f(x)} +f(1- x) =2x -x^4$ A function $f(x)$ satisfies the  functional equation $x^2{f(x)} +f(1- x) =2x -x^4$ for all real $x$. Then $f(x)$ is given by.
My work 
$$x^2{f(x)} +f(1- x) =2x -x^4$$
Replacing $x$ $by$ $1- x$ 
$$(1-x)^2{f(1 - x)} +f(x) =2x -x^4 + 4x^3 -6x^2 + 1$$
Subtracting them
$${f(x)}(1-x^2) +{f(1- x)}((1-x)^2 - 1) = 4x^3 -6x^2 + 1$$
What should I do next ?

You have the equation 
\begin{equation}
x^2\cdot f(x) + f(1-x) = 2x-x^4.
\end{equation}
Replacing $x$ by $1-x$ results in
\begin{equation}
(1-x)^2\cdot f(1-x) + f(x) = 2(1-x)-(1-x)^4.
\end{equation}
Now we have two equations with two unknowns, namely $f(x)$ and $f(1-x)$. We can subtract $(1-x)^2$ times the first equation from the second one to get
\begin{equation}
\left(1-(1-x)^2x^2\right)\cdot f(x) = 2(1-x)-(1-x)^4 - (1-x)^2\cdot (2x-x^4).
\end{equation}
This can be simplified to $$f(x)=1-x^2.$$
