\(x+y+z=3\Rightarrow x^2+y^2+z^2+2\left(xy+yz+zx\right)=9\Leftrightarrow xy+yz+zx=0\left(\text{vì:}x^2+y^2+z^2=9\right)\)
\(xy+yz+zx=0\Rightarrow xy=-yz-zx;yz=-xy-xz;xz=-xy-yz\)
\(P=\frac{-x\left(y+z\right)}{x^2}+\frac{-y\left(z+x\right)}{y^2}+\frac{-z\left(x+y\right)}{z}-4=\frac{y+z}{-x}+\frac{z+y}{-y}+\frac{x+y}{-z}-4\)
\(P=\frac{3}{x}+\frac{3}{y}+\frac{3}{z}-1=\frac{3yz+3xz+3xy}{xyz}-1=0-1=-1\)
Mk k hiểu dòng cuối
\(\frac{x+y}{-z}+\frac{y+z}{-x}+\frac{z+x}{-y}-4=\left(\frac{x+y}{-z}-1\right)+\left(\frac{y+z}{-x}-1\right)+\left(\frac{z+x}{-y}-1\right)-1\)
\(=\frac{x+y-\left(-z\right)}{-z}+\frac{y+z-\left(-x\right)}{-x}+\frac{z+x-\left(-y\right)}{-y}-1=\left(x+y+z\right)\left(\frac{1}{-x}+\frac{1}{-y}+\frac{1}{-z}\right)-1\)
\(=\frac{3}{-x}+\frac{3}{-y}+\frac{3}{-z}-1=\frac{-3xy-3yz-3zx}{xyz}-1=0-1=-1\)