Let'e rearrange the inequality

#6x^2+3x>=-5x-3+x^2#

#5x^2+8x+3>=0#

We factorise

#(5x+3)(x+1)>=0#

Let #f(x)=(5x+3)(x+1)#

We can build the sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-1##color(white)(aaaa)##-3/5##color(white)(aaaa)##+oo#

#color(white)(aaaa)##x+1##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#

#color(white)(aaaa)##5x+3##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#

#color(white)(aaaa)##f(x)##color(white)(aaaaaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#

Therefore,

#f(x>=0)# when #x in ]-oo,-1]uu [-3/5, +oo[#