\(1+xy=2\left(x^2+y^2\right)\ge4\left|xy\right|\ge4xy\)
\(\Rightarrow3xy\le1\Rightarrow xy\le\frac{1}{3}\)
\(1+xy\ge4\left|xy\right|\ge-4xy\Rightarrow5xy\ge-1\Rightarrow xy\ge-\frac{1}{5}\)
\(\Rightarrow-\frac{1}{5}\le xy\le\frac{1}{3}\)
\(P=7\left(x^4+y^4+2x^2y^2\right)-10x^2y^2=7\left(x^2+y^2\right)^2-10x^2y^2\)
\(P=\frac{7}{4}\left(xy+1\right)^2-10x^2y^2=-\frac{33}{4}x^2y^2+\frac{7}{2}xy+\frac{7}{4}\)
Đặt \(t=xy\Rightarrow P=f\left(t\right)=-\frac{33}{4}t^2+\frac{7}{2}t+\frac{7}{4}\) với \(t\in\left[-\frac{1}{5};\frac{1}{3}\right]\)
Xét \(f\left(t\right)\) trên \(\left[-\frac{1}{5};\frac{1}{3}\right]\)
\(f\left(-\frac{1}{5}\right)=\frac{18}{25}\) ; \(f\left(\frac{1}{3}\right)=2\) ; \(f\left(-\frac{b}{2a}\right)=f\left(\frac{7}{33}\right)=\frac{70}{33}\)
\(\Rightarrow M=\frac{70}{33}\) ; \(m=\frac{18}{25}\)