Given a polynomial 6x^4+2x^2-8x-c where c is a constant, for what value of c will make this expression have no remainder?

Let's try this:  Assume that one root of the polynomial is c.  Use this c as divisor in sythetic division:

     
c  / 6    0     2           -8                -c
           6c   6c^2      6c^3+2c      6c^4+2c^2-8c
   
      6   6c   6c^2+2  6c^3+2c-8   6c^4+2c^2-9c   Set 6c^4+2c^2-9c=0 
                                                                             (remainder is zero)
                                                                              and solve for c:  c=0
                                                                              is a solution.  

In other words, if c=0, the remainder obtained thru synth. div. will be zero.