Let's factorise the denominator
#x^2-5x+6=(x+1)(x-6)#
Let #f(x)=(x+3)/((x+1)(x-6))#
The domain of #f(x)# is #D_f(x)=RR-{-1,6}#
Now, we can build the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-3##color(white)(aaaaaa)##-1##color(white)(aaaaaaaaa)##6##color(white)(aaaaaa)##+oo#
#color(white)(aaaa)##x+3##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaa)##||##color(white)(aaaa)##+##color(white)(aaa)##||##color(white)(aaaa)##+#
#color(white)(aaaa)##x+1##color(white)(aaaaaa)##-##color(white)(aaaa)##-##color(white)(aaa)##||##color(white)(aaaa)##+##color(white)(aaa)##||##color(white)(aaaa)##+#
#color(white)(aaaa)##x-6##color(white)(aaaaaa)##-##color(white)(aaaa)##-##color(white)(aaa)##||##color(white)(aaaa)##-##color(white)(aaa)##||##color(white)(aaaa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaaa)##-##color(white)(aaaa)##+##color(white)(aaa)##||##color(white)(aaaa)##-##color(white)(aaa)##||##color(white)(aaaa)##+#
Therefore,
#f(x)<=0# when #x in ]-oo,-3] uu ]-1,6[#