$$ \begin{aligned} & \lim _{x \rightarrow 1} \frac{\left(\frac{\sin \left(3 x^{2}-4 x+1\right)}{3 x^{2}-4 x+1}\right)\left(3 x^{2}-4 x+1\right)-x^{2}+1}{2 x^{3}-7 x^{2}+a x+b}=-2 \\\\ \Rightarrow & \lim _{x \rightarrow 1} \frac{3 x^{2}-4 x+1-x^{2}+1}{2 x^{3}-7 x^{2}+a x+b}=-2 \\\\ \Rightarrow & \lim _{x \rightarrow 1} \frac{2(x-1)^{2}}{2 x^{3}-7 x^{2}+a x+b}=-2 \end{aligned} $$

So $f(x)=2 x^{3}-7 x^{2}+a x+b=0$ has $x=1$ as repeated root, therefore $f(1)=0$ and $f^{\prime}(1)=0$ gives

$$ a+b+5 \text { and } a=8 $$

So, $a-b=11$