<s>[INST] What is the simplest way to factor the following polynomial $x^4-8x^2+x+12$? 
What is the simplest  way  to factor the following  polynomial  $$x^4-8x^2+x+12$$ ?

Note : I already knew the factorization which is 
$$(x^2-x-3)(x^2+x-4)$$ 
I need the way to get that 
Thank you for your help.
 [/INST] We can write $$x^4-8x^2+x+12=(x^2+ax+b)(x^2+cx+d)$$
Comparing the coefficients of $x^3,$ $$0=a+c\iff c=-a$$
Comparing the coefficients of $x,$  $$1=ad+bc=a(d-b)$$
$$\implies a=d-b=\pm1$$
Comparing the constants,  $$bd=12$$
If $d-b=1, d=4,b=3$ or $d=-3.b=-4$
What if $d-b=-1?$
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