<s>[INST] Factorize $(x+1)(x+2)(x+3)(x+6)- 3x^2$ I'm preparing for an exam and was solving a few sample questions when I got this question -
Factorize : $$(x+1)(x+2)(x+3)(x+6)- 3x^2$$ 
I don't really know where to start, but I expanded everything to get :
$$x^4 + 12x^3 + 44x^2 + 72x + 36$$
I used rational roots test and Descartes rule of signs to get guesses for the roots. I tried them all and it appears that this polynomial has no rational roots.So, what  should I do to factorize this polynomial ?  
(I used wolfram alpha and got the factorization : $(x^2 + 4x + 6) (x^2 + 8x + 6)$ But can someone explain how to get there ?)  
 [/INST] $$(x+1)(x+2)(x+3)(x+6)-3x²$$
$$(x+1)(x+6)(x+3)(x+2)-3x²$$
$$(x²+6x+1x+6)(x²+3x+2x+6)-3x²$$
$$(x²+6x+6+1x)(x²+5x+6+x-x)-3x²$$
$$(x²+6x+6+1x)(x²+5x+x+6-x)-3x²$$
$$(x²+6x+6+1x)(x²+6x+6-x)-3x²$$
$$(x²+6x+6)²(+1x)(-x)-3x²$$
$$(x²+6x+6)²-x²-3x²$$
$$(x²+6x+6)²-4x²$$
$$(x²+6x+6)²-(2x)²$$
Apply $a²-b²=(a+b)(a-b)$
$$(x²+6x+6+2x)(x²+6x+6-2x)$$
$$(x²+8x+6)(x²+6x-2x+6)$$
$$(x²+8x+6)(x²+4x+6)$$
$$x(x+8+6/x)x(x+4+6/x)$$
Answer: $$x²(x+8+6/x)(x+4+6/x)$$
</s>