How do I write the function in the form f(x) = (x-k)q(x) + r(x) for the given value of k?

We are actually dividing f(x)  by the linear factor  (x -4)

This is analogous to testing to see if  4 is a root of  f(x)

So....we can perform synthetic division as follows

 4  [  15     10      - 6    0        14    ]

                 60     280  1096   4384

      ________________________

       15     70     274  1096    4398

So  q(x)  =   15x^3 + 70x^2 + 274x + 1096    

And   r(x)   =   4398 / [ x - 4 ]