\(\left(x+y\right)^2+xy^2+2y^3=9y^2+8x\)
\(\Leftrightarrow x^2+y^2+2xy+xy^2+2y^3=9y^2+8x\)
\(\Leftrightarrow xy^2+x^2-8y^2-8x+2xy+2y^3=0\)
\(\Leftrightarrow x\left(y^2+x\right)-8\left(y^2+x\right)+2y\left(y^2+x\right)=0\)
\(\Leftrightarrow\left(y^2+x\right)\left(x-8+2y\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}y^2+x=0\\x+2y=8\end{matrix}\right.\)
TH1: \(y^2+x=0\Leftrightarrow x=y=0\), thỏa mãn.
TH2: \(x+2y=8\Rightarrow\left(x;y\right)\in\left\{\left(0;4\right);\left(2;3\right);\left(4;2\right);\left(6;1\right);\left(8;0\right)\right\}\)
Vậy pt đã cho có các cặp nghiệm tự nhiên (x; y) là:
\(\left(x;y\right)\in\left\{\left(0;0\right);\left(0;4\right);\left(2;3\right);\left(4;2\right);\left(6;1\right);\left(8;0\right)\right\}\)
