Working with Vieta

Algebra Level 5

Let $a,b,c$ be the real roots of the cubic equation $2t^3+3t^2-9t-6=0$ . Suppose that $\left|\left(\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}\right)\left(\dfrac{b-c}{a}+\dfrac{c-a}{b}+\dfrac{a-b}{c}\right)\right|=\dfrac{m}{n}$ for some relatively prime positive integers $m$ and $n$ . Find $m+n$ .