Let #f(x)=(x^2+5x)/(x-3)=(x(x+5))/(x-3)#
The domain of #f(x)# is #D_(f(x))=RR-{3}#
Let's do a sign chart to solve this inequality
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-5##color(white)(aaaa)##0##color(white)(aaaa)##3##color(white)(aaaa)##+oo#
#color(white)(aaaa)##x+5##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaa)##+##color(white)(aaa)##+#
#color(white)(aaaa)##x##color(white)(aaaaaaaaa)##-##color(white)(aaaa)##-##color(white)(aa)##+##color(white)(aaa)##+#
#color(white)(aaaa)##x-3##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaa)##-##color(white)(aaa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaa)##-##color(white)(aaa)##+#
Therefore #f(x)>=0# when #x in [-5,0 ] uu ]3, +oo[#
graph{(x^2+5x)/(x-3) [-72.6, 75.47, -19.23, 54.86]}