Let's rewrite the inequality
#x^2-3x-54>0#
Let's factorise
#x^2-3x-54=(x+6)(x-9)#
and let #f(x)=x^2-3x-54#
Now we can make the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-6##color(white)(aaaa)##9##color(white)(aaaaa)##+oo#
#color(white)(aaaa)##x+6##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x-9##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#
Therefore,
#f(x)>0#, when # x in ] -oo,-6 [ uu ] 9, oo [#