<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] 
Another linked question is to factorise $\,f(x) = (x + 1)(x + 3)(x + 5)(x + 7) + 15\,$

Let $z=x+4$ then:
$$
\begin{align}
(z-3)(z-1)(z+1)(z+3)+15 & = (z^2-1)(z^2-9)+15=z^4-10z^2+24 \\
 & = (z^2-4)(z^2-6) = (x^2+8x+12)(x^2+8x+10) \\
 & = (x+2)(x+6)(x^2+8x+10)
\end{align}
$$
</s>