Fast way to come up with solutions to $x(x-1)(x-2)(x-3)=1$?

Question

I can solve this equation $x(x-1)(x-2)(x-3)=1$ using the usual method but I am looking for a fast analytical method to solve this. Any hints ?

Answer 1

HINT:

$x(x-3)=x^2-3x, (x-1)(x-2)=x^2-3x+2$

Write $x^2-3x=u$

Answer 2

Let $y=x-{3 \over 2}$ to get $y^4-{ 5 \over 2} y^2 - {7 \over 16}$.

(A plot of the function shows a local maximum at ${3\over 2}$.)