Explanation:
ok, so
(3x³ + 4x² + 8) ÷ (x + 2) =
remember, that is similar to number division. what we do first ?
we take the left term of the dividend 3x³ and divide by the left term of the divisor x :
3x³ ÷ x = 3x²
that result becomes our first term of the quotient (result).
(3x³ + 4x² + 8) ÷ (x + 2) = 3x²
now we multiply the dividend by this term and subtract it from the left part of the dividend, and then we pull down the next term from the original dividend :
(3x³ + 4x² + 8) ÷ (x + 2) = 3x²
- 3x³ + 6x²
-----------------
0 -2x²
+ 8
and we repeat this with the new dividend.
the left term of the dividend -2x² is divided by the left term of the divisor x.
-2x² ÷ x = -2x
that result becomes our next term of the quotient.
(3x³ + 4x² + 8) ÷ (x + 2) = 3x² - 2x
- 3x³ + 6x²
-----------------
0 -2x²
+ 8
----------------------
-2x² + 8
and again we multiply the divisor by this new term and subtract that result from the latest dividend. and if there would be another term in the original dividend, we would pull it down.
(3x³ + 4x² + 8) ÷ (x + 2) = 3x² - 2x
- 3x³ + 6x²
-----------------
0 -2x²
+ 8
----------------------
-2x² + 8
- -2x² - 4x
------------------------
0 + 4x + 8
in this case we could not pull down a new term, but there is enough remainder to do another pass the same way as before.
we divide the left term of the actual dividend 4x by the left term of the divisor x.
4x ÷ x = 4
that result becomes our next term of the quotient.
(3x³ + 4x² + 8) ÷ (x + 2) = 3x² - 2x + 4
- 3x³ + 6x²
-----------------
0 -2x²
+ 8
----------------------
-2x² + 8
- -2x² - 4x
------------------------
0 + 4x + 8
and again, we multiply the divisor by the latest term and subtract this result from the left part of the actual dividend, and again, if there is still something to pull down from the original dividend, we would pull that down.
(3x³ + 4x² + 8) ÷ (x + 2) = 3x² - 2x + 4
- 3x³ + 6x²
-----------------
0 -2x²
+ 8
----------------------
-2x² + 8
- -2x² - 4x
------------------------
0 + 4x + 8
- 4x + 8
-----------------------------
0 0
there is no remainder and no more terms from the original dividend to pull down, so, we are finished.
the quotient (result) is
(3x³ + 4x² +8) ÷ (x + 2) = 3x² - 2x + 4
FYI - if there would have been a remainder (like a constant c), then c/(x + 2) would have been added to the result.