When dividing the polynomials by synthetic division,
what are the numbers we list across the top row in 
the division? 



You will notice there is a skip in the exponents of x
in the top. That is, there is no term in x³ so we must
insert the term +0x³:

 


Then the top line of the synthetic division goes:

        2  0 -1  5 -7 

To do the synthetic division:

     -4|2  0 -1    5  -7
       |  -8 32 -124 476
        2 -8 31 -119 469

The division gives

      

and is a shortcut for this long division:

               2x³ -  8x² +  31x - 119        
   x + 4)2x4 + 0x³ -   x² +   5x -   7
         2x4 + 8x³
              -8x³ -   x²
              -8x² - 32x²   
                     31x² +   5x
                     31x² + 124x
                           -119x -   7
                           -119x - 476 
                                   469

which gives the exact same answer, because if you will
notice all the coefficients in the long division also 
occur as numbers in the synthetic division, except 
that the signs of the subtrahends in the long division
are opposite because subtraction is done in the long 
divion whereas assition is done in the synthetic division.
This conversion from subtraction to addition is 
accomplished by changing the sign of the 4 in the
divisor x+4 to -4 in the synthetic division. 

Edwin