How is the remainder theorem used to find the remainder of (4x^4-2x^3-52x^2-15x)/(x-4)? find the remainder.

One way to do this would be to use synthetic division.  Use 4 as the divisor and use the coefficients 4, -2, -52, -15, 0  (yes, the last coeff. is a zero):

   
4 /  4   -2   -52   -15   0
          16    56     16   4
   
       4  14     4     1      4          The remainder is 4.  That means that if you divide the given 4th order polynomial by (x-4), the remainder is 4.

Note also, that the value of the polynomial, when x = 4, is 4.