Show your work for factoring or quadratic formula: 2x^4 - 8x^3 +10x^2 - 8x + 8 = 0 , and possibly show the work.

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