Look at the bottom first.
4x^2 - 4
4(x^2 - 1)
4(x-1)(x+1)
So the bottom factors to 4, x-1, and x+1
The top can't factor out a 4 since there's no 4 on the x^3.
If we could factor x+1 out of the top, then plugging in -1 for x on top would make the top zero. Let's try that.
(-1)^3 + 4(-1)^2 + (-1) - 6
-1 + 4 - 1 - 6
-4
No. x = -1 doesn't make the top zero, so x+1 won't factor out of the top.
If x-1 factors out of the top then x = 1 would make the top zero. Let's try that.
1^3 + 4(1)^2 + 1 - 6
1 + 4 + 1 - 6
0
x = 1 makes the top zero, so x-1 should factor out.
If you use synthetic division or some other method, you'll find that x^3 + 4x^2 + x - 6 = (x-1)(x^2 + 5x + 6)
(x^3 + 4x^2 + x - 6) / 4(x - 1)(x + 1)
(x - 1)(x^2 + 5x + 6) / 4(x - 1)(x + 1)
(x^2 + 5x + 6) / 4(x+1)
The top factors to. . .
(x + 2)(x + 3) / 4(x + 1)
Nothing else cancels out, so that's it.