<s>[INST] Factorise $\,a(a - 4)(a^2 - 4a - 1) - 20\,$ I am making an attempt to factorise the above equation.
To do this i am expanding the already factored part of the above equation to obtain ->
$\,f(a) = a^4 - 8a^3 + 15a^2 + 4a - 20\,$
Now i do some random guesses to find that f(2) = 0, hence i know that  (a-2) will divide f(a) so i perform long division to get f(a) = (a-2)( Something ) and so on...  until i get - >  $\,f(a) = (a - 2)^2(a + 1)(a - 5)\,$        
I feel that i am approaching the problem in an incorrect and long way. Please let me know if i can get the factorised form in a cleaner and faster way.
PS: Another linked question is to factorise $\,f(x) = (x + 1)(x + 3)(x + 5)(x + 7) + 15\,$   
 [/INST] $1)$ Call $x=a(a-4)$ then
$$x(x-1)-20=x^2-x-20=(x+4)(x-5)$$
then
$$(a^2-4a+4)(a^2-4a-5)=(a-2)^2(a+1)(a-5)$$
$2)$ For
$$(x+1)(x+3)(x+5)(x+7)+15=[(x+3)(x+5)][(x+1)(x+7)]+15=(x^2+8x+15)(x^2+8x+7)+15$$
Call $y=x^2+8x+7$ then:
$$(y+8)y+15=y^2+8y+15=(y+3)(y+5)$$
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