<s>[INST] Formal way of showing inequality? At the end of my class today, there was a proof that involved using the fact that given $f(x)=3+4x-x^2-x^4$ 
$f(x)\leqslant 0 $ if $|x|>2 $
this seems obvious but I was wondering if anyone could show me a more formal way of proving it?
 [/INST] For $x\geq2$ we have $$x^4+x^2-4x-3=x^4-2x^3+2x^3-4x^2+3x^2-6x+2x-4+1=$$
$$=(x-2)(x^3+2x^2+3x+2)+1>0$$
For $x\leq-2$ we obtain:
$$x^4+x^2-4x-3=x^4+3x^3+4x^2-4x^3-16x^2-16x+13x^2+52x+52-40x-80+25=$$
$$=(x+2)^2(x^2-4x+13)-40(x+2)+25>0$$
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