To use synthetic division to divide x^2 + 2x - 4 by x - 2, we set up the following synthetic division table:
2 | 1 2 -4
|___ 6
| 1 8 2
The first row of the table contains the coefficients of the quadratic polynomial, written in descending order of degree. The number 2 in the leftmost column of the table is the divisor, x - 2, written with the opposite sign.
To start the division, we bring down the first coefficient, 1, to the bottom row of the table.
Next, we multiply the divisor, 2, by the number in the bottom row, 1, and write the result in the second row, under the coefficient of x:
2 times 1 is 2, so we write 2 in the second row, under the 2.
We then add the numbers in the second row (6) and the second column (2), and write the result in the third row, under the coefficient of the constant term:
6 + 2 = 8, so we write 8 in the third row, under the -4.
The numbers in the bottom row of the table represent the coefficients of the quotient polynomial, and the number in the rightmost cell of the table represents the remainder.
Therefore, we have:
x^2 + 2x - 4 = (x - 2)(x + 6) + 8
or equivalently,
x^2 + 2x - 4 = (x - 2)(x + 6) - 8/(x-2)