\(\dfrac{a}{a^2+7}=\dfrac{a}{ab+bc+ca+a^2}=\dfrac{a}{\left(a+b\right)\left(a+c\right)}\)
Cmtt: \(\left\{{}\begin{matrix}\dfrac{b}{b^2+7}=\dfrac{b}{\left(a+b\right)\left(b+c\right)}\\\dfrac{c}{c^2+7}=\dfrac{c}{\left(a+c\right)\left(b+c\right)}\end{matrix}\right.\)
Do đó \(VT=\dfrac{a}{\left(a+b\right)\left(a+c\right)}+\dfrac{b}{\left(a+b\right)\left(b+c\right)}+\dfrac{c}{\left(a+c\right)\left(b+c\right)}\)
\(=\dfrac{a\left(b+c\right)+b\left(a+c\right)+c\left(a+b\right)}{\left(a+b\right)\left(b+c\right)\left(a+c\right)}\\ =\dfrac{ab+ac+ac+bc+ac+bc}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\\ =\dfrac{2\left(ab+bc+ca\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=\dfrac{2\cdot7}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\\ =\dfrac{14}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)
