\(A=\frac{x^2\left(x-1\right)+y^2\left(y-1\right)}{\left(x-1\right)\left(y-1\right)}=\frac{x^3+y^3-\left(x^2+y^2\right)}{\left(x-1\right)\left(y-1\right)}\)
Đặt \(\left\{{}\begin{matrix}x-1=a>0\\y-1=b>0\end{matrix}\right.\)
\(A=\frac{\left(a+1\right)^2}{b}+\frac{\left(b+1\right)^2}{a}=\frac{a^2+2a+1}{b}+\frac{b^2+2b+1}{a}\)
\(=\frac{a^2}{b}+\frac{b^2}{a}+\frac{1}{a}+\frac{1}{b}+2\left(\frac{a}{b}+\frac{b}{a}\right)\)
\(A\ge\frac{\left(a+b\right)^2}{a+b}+\frac{4}{a+b}+4=a+b+\frac{4}{a+b}+4\ge2\sqrt{\frac{4\left(a+b\right)}{a+b}}+4=8\)
\(A_{min}=8\) khi \(a=b=1\) hay \(x=y=2\)
