jmc

The inequality $x^2-5x+6<0$ factors as $(x-2)(x-3)<0,$ so the solution is $2<x<3.$ Since $x^2+5x+6$ is increasing on this interval, we have that $$x^2+5x+6>2^2+5\cdot 2+6=20$$and $$x^2+5x+6<3^2+5\cdot 3+6=30.$$Therefore, the set of possible values of $x^2+5x+6$ is $\boxed{(20,30)}.$