Problem:  Divide  
x4 - 2x3 + 8x - 14
x2 - 3


Set up the long division.
Notice the 0's put in as place
holders for missing powers of x.
x2 =   x4
x2
.
Choose x2 since
x2 · x2 matches x4.
x4 + 0x3- 3x2 = x2(x2 + 0x - 3).
Subtract x4 + 0x3 - 3x2 from
x4 - 2x3 + 0x2 + 8x - 14.
Result is -2x3 + 3x2 + 8x - 14.
-2x =   -2x3
x2
.
Choose -2x since
-2x · x2 matches -2x3.
-2x3 + 0x2 + 6x = -2x(x2 + 0x - 3).
Subtract -2x3 + 0x2 + 6x from
-2x3 + 3x2 + 8x - 14..
Result is 3x2 + 2x - 14.
3 =   3x2
x2
.
Choose 3 since
3 · x2 matches 3x2.
3x2 + 0x2 - 9 = 3(x2 + 0x - 3).
Subtract 3x2 + 0x2 - 9 from
3x2 + 2x - 14..
Result is 2x - 5.
2x - 5 is the remainder.


Answer:   x4 - 2x3 + 8x - 14
x2 - 3
 =   x2 - 2x + 3 + 2x - 5
x2 - 3
.


Notes
x4 - 2x3 + 8x - 14 = (x2 - 3)(x2 - 2x + 3) + 2x - 5
Dividend x4 - 2x3 + 8x - 14
Divisor x2 - 3
Quotient x2 - 2x + 3
Remainder 2x - 5