<s>[INST] Find all real functions $f$ such that $f(xf(y))=(1-y)f(xy)+x^2y^2f(y)$ 
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x,y$ we have:
  $f(xf(y))=(1-y)f(xy)+x^2y^2f(y)$

I have solved this problem but the solution is "bruteforce", so I wanted to ask if there is a more elegant way of approaching this.
 [/INST] I claim that the only possible solutions to the functional equation
$$f\big(xf(y)\big)=(1-y)f(xy)+x^2y^2f(y)\tag0\label0$$
are $f(x)=0$ and $f(x)=x-x^2$.
To show that, first let $y=1$ in \eqref{0} and you'll get $f\big(xf(1)\big)=x^2f(1)$. Now assuming $f(1)\neq0$ we'll have $f(x)=\frac{x^2}{f(1)}$ for every real number $x$, whcih leads to a contradiction using \eqref{0}. So we have:
$$f(1)=0\tag1\label1$$
Next, let $x=1$ in \eqref{0} and you'll get:
$$f\big(f(y)\big)=\big(1-y+y^2\big)f(y)\tag2\label2$$
Now, substituting $f(y)$ for $y$ in \eqref{0} and using \eqref{2} we get:
$$f\Big(xf\big(f(y)\big)\Big)=\big(1-f(y)\big)f\big(xf(y)\big)+x^2f(y)^2f\big(f(y)\big)\\
\therefore f\Big(x\big(1-y+y^2\big)f(y)\Big)=\big(1-f(y)\big)f\big(xf(y)\big)+x^2f(y)^3\big(1-y+y^2\big)$$
So, if $f(y)\neq0$, letting $x=\frac1{f(y)}$ in the last equation and using \eqref{1}, we get:
$$f\big(1-y+y^2\big)=\big(1-y+y^2\big)f(y)\ne0$$
Thus by \eqref{2}, $f\big(f(y)\big)=f\big(1-y+y^2\big)\ne0$ and so $f\Big(f\big(f(y)\big)\Big)=f\Big(f\big(1-y+y^2\big)\Big)$. Hence by \eqref{2}:
$$\big(1-f(y)+f(y)^2\big)f\big(f(y)\big)=\Big(1-\big(1-y+y^2\big)+\big(1-y+y^2\big)^2\Big)f\big(1-y+y^2\big)\\
\therefore1-f(y)+f(y)^2=1-\big(1-y+y^2\big)+\big(1-y+y^2\big)^2\\
\therefore\Big(f(y)-\big(1-y+y^2\big)\Big)\Big(f(y)+\big(1-y+y^2\big)-1\Big)=0$$
$$\therefore f(y)=1-y+y^2\quad or\quad f(y)=y-y^2\tag3\label3$$
Now, if $f(y)=1-y+y^2$ then $f\big(1-y+y^2\big)=\big(1-y+y^2\big)^2$, but because $1-y+y^2\neq0$ we must have $f\big(1-y+y^2\big)=1-\big(1-y+y^2\big)+\big(1-y+y^2\big)^2$ or $f\big(1-y+y^2\big)=\big(1-y+y^2\big)-\big(1-y+y^2\big)^2$ by \eqref{3}. It's easy to check that the latter case leads to a contradiction and the former case leads to $y=0$ or $y=1$. But letting $x=y=0$ in \eqref{0} we have $f(0)=0$, and by \eqref{1} we have $f(1)=0$. So this case leads to a contradiction, too, since we assumed $f(y)\neq0$. So $f(y)$ cannot be equal to $1-y+y^2$ and hence by \eqref{3}, $f(y)=y-y^2$.
Finally, if there is a real number $y$ such that $y\neq0$, $y\neq1$ and $f(y)=0$, then by \eqref{0} we get $f(0)=(1-y)f(xy)+0$ for every real number $x$, which means that $f$ is the constant zero function. This yields our first claim. It's easy to check that indeed the two mentioned functions satisfy \eqref{0}.
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