The equation of common tangent to the curves $y=6-x-x^2$  and $xy=x+3$  is  
 

Let $y=mx+c$   is tangent to $y=6-x-x^2$   and $xy=x+3$   then 
$mx+c=6-x-x^2$   and    $x(mx+c)=x+3$
equations have equal roots ${(m+1)}^2=4(c-6)$  and    ${(c-1)}^2=-12m$
By  solving   $m=-3,c=7$   $\therefore y=-3x+7$