If m is slope of common tangents y=$x^2-x+1$ , $y=x^2-3x+1$ then m is

A point of first curve is $P(t_1,t_1{}^2-t+1)$ , slope of tangent at $P=2t_1-1$ . So, the equation of tangent at P is  

$y-(t_1{}^2-t+1)=(2t_1-1)(x-t_1)\mathrm{.......}\left(1\right)$

A point  on second curve is  $Q(t_2,t_2{}^2-3t_2+1)$ , slope of tangent at  $Q=2t_2-3$

So, the equation of tangent at Q is

$y-(t_2{}^2-3t+1)=(2t_2-3)(x-t_2)\mathrm{..........}\left(2\right)$

and (2) are same lines, so,

$2t_1-1=2t_2-3$   $\Rightarrow t_2-t_1=1$ …………..(4)

And

$-t_1(2t_1-1)+t_1{}^2-t+1=-t_2(2t_2-3)+t_2{}^2-3t_2+1$

$\Rightarrow -t_1{}^2+1=-t_2{}^2+1$

$t_1=\pm t_2$ $\Rightarrow t_2=-t_1$ $(\because t_1\ne t_2)$

$\therefore From\left(4\right),t_1=-\frac{1}{2}$

Desired slope =$2t_1-1=-2$