\(\left(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}\right)\left(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c}\right)\)
\(=3+\frac{a}{b-c}\left(\frac{c-a}{b}+\frac{a-b}{c}\right)+\frac{b}{c-a}\left(\frac{b-c}{a}+\frac{a-b}{c}\right)+\frac{c}{a-b}\left(\frac{b-c}{a}+\frac{c-a}{b}\right)\)
\(=3+\frac{a}{b-c}.\frac{c^2-ac+ab-b^2}{bc}+\frac{b}{c-a}.\frac{bc-c^2+a^2-ab}{ac}+\frac{c}{a-b}.\frac{b^2-bc+ac-a^2}{ab}\)
\(=3+\frac{a\left(b-c\right)\left(a-b-c\right)}{\left(b-c\right)bc}+\frac{b\left(c-a\right)\left(b-c-a\right)}{\left(c-a\right)ac}+\frac{c\left(a-b\right)\left(c-a-b\right)}{\left(a-b\right)ab}\)
\(=3+\frac{2a^2}{bc}+\frac{2b^2}{ac}+\frac{2c^2}{ab}\)
\(=3+2.\frac{a^3+b^3+c^3}{abc}\)
\(=3+2.\frac{\left(a+b+c\right)^2-3\left(a+b\right)\left(b+c\right)\left(c+a\right)}{abc}\)
\(=3+2.\frac{0+3abc}{abc}\)
\(=9\left(đpcm\right)\)
