How to find the quotient using polynomial long division?

You can follow the same steps as integer division.

. ._____
2 ) 510

What are the steps? divide, multiply, subtract, continue. So here, 5 / 2 = 2, 2 * 2 = 4, 5 - 4 = 1, then drop down the other number:

. ._2____
2 ) 510
. . .4
. . ---
. . .11

Repeating: 11 / 2 = 5, 5 * 2 = 10, 11 - 10 = 1, drop down the next number:

. ._25__
2 ) 510
. . .4
. . ---
. . .11
. . .10
. . ------
. . . .10

And the last digit is 5, for the answer of 255.

In your polynomial, same steps:

. . . . . _____________
2x + 1 ) 2x² - 5x - 3

divide: 2x² / 2x = x, multiply: x * (2x + 1) = 2x² + x, then subtract:

. . . . . __x___________
2x + 1 ) 2x² - 5x - 3
. . . . . . 2x² + x
. . . . . ----------------
. . . . . . . . . -6x - 3

And one more, with -3:

. . . . . __x_-_3______
2x + 1 ) 2x² - 5x - 3
. . . . . . 2x² + x
. . . . . ----------------
. . . . . . . . . -6x - 3
. . . . . . . . . -6x - 3
. . . . . . . . --------------
. . . . . . . . . . . . . 0

No remainder, so the answer is:

x - 3

If there was a reminder, again, recall what you do with integers:

5 / 2 = 2 R 1

The remainder is a numerator with the denominator equal to the divisor. So:

5 / 2 = 2 1/2

So in the above polynomial case, the divisor was (2x + 1), so if remainder was 1, we'd have to add 1/(2x + 1) to the answer.

Here's a pic of the division:
http://imgur.com/9lBqk3J

Well,

2x^2 - 5x - 3 | 2x+1
-2x^2 - x.......... x - 3
___________
..........-6x - 3
...........6x + 3
___________
............. 0
therefore :
2x^2 - 5x - 3 = (2x + 1)(x - 3)

hope it' ll help !!