Finding the zeros i do not understand what the next step would be, since the equation produced from synthetic division is not factorable. the professor provided the answers we should get from performing the correct steps. any body willing to me out here

Use the quadratic formula to solve the equations . you will get the answers 

Regarding #6:

You did well in identifying the possible rational roots.  Like you, I checked out x=3 as a possible root.  The 3 coefficients remaining after that are 1, -4 and 2.

They correspond to the quadratic expression 1x^2 -4x + 2.

We can apply the quadratic formula here.  a=1, b=-4 and c=2.

                  -(-4) plus or minus sqrt( [-4]^2 - 4(1)(2)  )
Then x =
                                              2(1)
              4 plus or minus sqrt(8)           4 plus or minus 2sqrt(2)
or  x =  =  
                            2                                                   2

Then x = 2 plus or minus sqrt(2).

If you wanted, you could check your work here by using synthetic division.  Let the divisor be 2+sqrt(2) and the 3 coefficients    be   1   -4   2.

The remainder should be zero.