Suppose for the moment that the inequality holds for all [tex]a,b[/tex]:
[tex](a-b)^2<(a+b)(a-b)<(a+b)^2[/tex]
Expanding everything gives
[tex]a^2-2ab+b^2<a^2-b^2<a^2+2ab+b^2[/tex]
[tex]\implies-ab<-b^2<ab[/tex]
In particular, the inequality says that [tex]-ab<ab[/tex] for any choice of [tex]a,b[/tex]. But if [tex]a<0[/tex] and [tex]b>0[/tex], then [tex]-ab>0[/tex] while [tex]ab<0[/tex].
So one possible choice of [tex]a,b[/tex] could be [tex]a=-1[/tex] and [tex]b=1[/tex]. Then we get
[tex](-1-1)^2=4[/tex]
[tex](-1+1)(-1-1)=2[/tex]
[tex](-1+1)^2=0[/tex]
but clearly it's not true that [tex]4<2<0[/tex].
