Algebraic Long Division

A Quadratic Expression
Divided By A Linear Expression

x² + 2x + 1       divided by      x + 1

                      x + 1
x + 1 ) x² + 2x + 1
           x² + 1x
                   x + 1
                   x + 1
                   0 + 0

x² + 2x + 1 = (x + 1)(x + 1) = (x + 1)²

A Cubic Expression
Divided By A Linear Expression

x³ + 2x² + 2x + 1       divided by      x + 1

                      x² +   x + 1
x + 1 ) x³ + 2x² + 2x + 1
             x³ + 1x²
                      x² + 2x
                      x² +   x
                                x + 1
                                x + 1
                                0 + 0

x³ + 2x² + 2x + 1 = (x² +   x + 1)(x + 1)

This will not factorise further

A Cubic Expression Without An x² Term
Divided By A Linear Expression

3x³ + 2x + 1       divided by      x + 1

REWRITE AS

3x³ + 0x² + 2x + 1       divided by      x + 1

                      3x² - 3x + 5 remainder - 4
x + 1 )3x³ + 0x² + 2x + 1
            3x³ + 3x²
                  - 3x² + 2x
                  - 3x² - 3x
                    + 5x + 1
                      + 5x + 5
                          0 - 4

x + 1 is not a factor of 3x³ + 2x + 1
because 3x³ + 2x + 1 does not divide by  x + 1
without leaving a remainder