(8x⁴ + 5x³ + 4x² - x + 7)/(x + 1)

For this problem, where the leading coefficient of the divisor is 1, plain 'ol long division will suffice here. 

           _____________________ 
 x + 1  |8x⁴ + 5x³ + 4x² - x + 7

So x + 1 goes into the dividend how many times? Ask yourself, what do I need to multiply x by in order to get back an 8x⁴?  8x³. So multiply 8x³ by the divisor like you would in regular long division, except when you're finished change the signs of those two terms (to account for the fact that we are really doing a subtraction) and sum together.


         ______8x³___________
x + 1 |8x⁴ + 5x³ + 4x² - x + 7
      (-)8x⁴ (-) 8x³
           0x⁴ - 3x³ + 4x²

Bring down the 4x² term and repeat. What do I need to multiply x by in order to get negative 3x³? -3x². Multiply by the divisor and change the signs (again, shown in parentheses) and sum.

         ______8x³ - 3x²_+ 7x_- 8
x + 1 |8x⁴ + 5x³ + 4x² - x + 7
      (-)8x⁴ (-)8x³
          0x⁴ - 3x³ + 4x²
              (+)3x³ (+)3x²
                  0x³  + 7x²  - x
                         (-)7x² (-)7x
                           0x² - 8x + 7
                               (+)8x (+)8
                                     0x + 15

This gives a remainder of 15, and like regular long division, can be written over the divisor as 15 / (x + 1)

So the answer is 8x³ - 3x² + 7x - 8 + (15/(x+1))