Factorize $a^2x(2−x)[2−ax(2−x)]−x=0$

Question

I'm trying to factorise this expression:

$a^2x(2-x)[2-ax(2-x)] - x = 0$

I'm aware that: $a^2x(1-x)[1-ax(1-x)]-x=0$, factorises to: $x[ax-(a-1)][a^2x^2-a(a+1)x+(a+1)]=0$,

but I can't achieve a similar looking result, any suggestions?

Answer 1

Expanding and factoring or even cancelling terms can be tedious.
I trust we already know about theory here so it's OK to use tools to cheat.

When in doubt you can expand terms with Wolfram Alpha and then turn around and factor the result.

$$a^2x(2-x)[2-ax(2-x)] - x\\ =-a^3 x^4 + 4 a^3 x^3 - 4 a^3 x^2 - 2 a^2 x^2 + 4 a^2 x - x\\ =-x (a x - 2 a + 1) (a^2 x^2 - 2 a^2 x - a x + 2 a + 1) = 0$$

or you can factor the original expression as shown here.

$$-x (a x - 2 a + 1) (a^2 x^2 - 2 a^2 x - a x + 2 a + 1)=0$$