$${Find}\:{all}\:{functions}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:{such}\:{that} \\$$$${f}\left({x}+{y}\right)=\mathrm{2}{f}\left({x}\right)+\mathrm{3}{f}\left({y}\right)−\mathrm{4}{xyf}\left(\mathrm{2}{x}−\mathrm{3}{y}\right) \\$$$$\left(\forall{x};{y}\in\mathbb{R}\right) \\$$
$${f}\left({x}\right)=\mathrm{5}{f}\left(\frac{{x}}{\mathrm{2}}\right)−{x}^{\mathrm{2}} {f}\left(−\frac{{x}}{\mathrm{2}}\right) \\$$$${also} \\$$$${f}\left({x}\right)=\mathrm{2}{f}\left({x}\right)+\mathrm{3}{f}\left(\mathrm{0}\right) \\$$$${and} \\$$$${f}\left(\mathrm{0}\right)=\mathrm{5}{f}\left(\mathrm{0}\right)\:\:\Rightarrow\:\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\$$$$\Rightarrow\:\:{f}\left({x}\right)=\mathrm{0} \\$$$${and} \\$$$${f}\left(\mathrm{2}{x}\right)=\mathrm{5}{f}\left({x}\right)−\mathrm{4}{x}^{\mathrm{2}} {f}\left(−{x}\right) \\$$$${f}\left(−\mathrm{2}{x}\right)=\mathrm{5}{f}\left(−{x}\right)−\mathrm{4}{x}^{\mathrm{2}} {f}\left({x}\right) \\$$$${say}\:\:{f}\left(\mathrm{2}{x}\right)={A},\:{f}\left(−\mathrm{2}{x}\right)={B} \\$$$${f}\left({x}\right)=\frac{\mathrm{5}{A}+\mathrm{4}{x}^{\mathrm{2}} {B}}{\mathrm{25}−\mathrm{16}{x}^{\mathrm{4}} } \\$$$${f}\left(−{x}\right)=\frac{\mathrm{4}{x}^{\mathrm{2}} {A}+\mathrm{5}{B}}{\mathrm{25}−\mathrm{16}{x}^{\mathrm{4}} } \\$$$$\Rightarrow\:\:{f}\left({x}\right)={f}\left(−{x}\right) \\$$$$\Rightarrow\:\:{f}\left(\mathrm{2}{x}\right)=\left(\mathrm{5}−\mathrm{4}{x}^{\mathrm{2}} \right){f}\left({x}\right) \\$$$$…….. \\$$