First, I'd simplify this problem by dividing every term by 2:
x^4 - 4x^3 + 5x^2 - 4x + 4 = 0
Then I'd think of the possible factors of that constant term, 4: They would be plus or minus 1, plus or minus 2, or plus or minus 4.
Let's try x = -4 as a possible root. Setting up and using synthetic division:
____________________
-4 / 1 -4 5 -4 4
-4 32
----------------------------
1 -8 37
This result is not helpful. So, try the possible root +4 instead:
________________
+4 / 1 -4 5 -4 4
4 0 20 64
----------------------------
1 0 5 16 68 4 is not a root because the remainder
(68) is not zero.
Try x = 2:
________________
+2 / 1 -4 5 -4 4
2 -4 2 -40
----------------------------
1 -2 1 -2 0
2 is a root because the remainder is zero.
Continue this process until you've found the other 3 roots.
Hint: Try the possible root x = -2 with the following coefficients:
____________
-2 / 1 -2 1 -2
---------------------