Divide 2x^3+5x^2+7x+1 by 2x+1?

2x³ + 5x² + 7x + 1
= (2x³ + x²) + 4x² + 7x + 1
= x²(2x + 1) + (4x² + 2x) + 5x + 1
= x²(2x + 1) + 2x(2x + 1) + [5x + (5/2)] - (3/2)
= x²(2x + 1) + 2x(2x + 1) + (5/2)(2x + 1) - (3/2)
= (2x + 1)[x² + 2x + (5/2)] - (3/2)

Hence, (2x³ + 5x² + 7x + 1) / (2x + 1)
= {(2x + 1)[x² + 2x + (5/2)] - (3/2)} / (2x + 1)
= x² + 2x + (5/2) - {3/[2(2x+ 1)]}

The answer: the first option

.............x^2 + 2x + 5/2 answer//
......... .__________________
2x + 1|2x^3 + 5x^2 + 7x + 1
..........-.2x^3 + x^2
....-------------------------------------
................. 4x^2 + 7x
..................4x^2 + 2x
...........-----------------------
......................... 5x + 1
..........................5x + 5/2
........................------------------
................................- 3/2

so
............................3
= x^2 + 2x - ------------------- + 5/2 answer//
..........................2(2x + 1)

Quotient and remainder:
2 x^3 + 5 x^2 + 7 x + 1 = (x^2 + 2 x + 5/2) × (2 x + 1) + -3/2

.... ..x^2+2x+ 5/2
----- --- -- -- - -- -
2x+1 ) 2x^3+5x^2+7x+1
.......2x^3+x^2
------ ----- ---------
.... .... ..4x^2+7x
..... ......4x^2+2x
------ ----- ---- ------
..... .. ........5x+1
.... .... .......5x+5/2
----------- -------- ---
..... ..... .......-3/2

Quotient = x^2+2x+5/2
remainder = -3/2

x^2+2x+5/2 -3/(2(2x+1))

The first choice

2x^3 + 5x^2 + 7x + 1
= 2x^3 + x^2 + 4x^2 + 7x + 1
= 2x^2(2x + 1) + 4x^2 + 2x + 5x + 1
= 2x^2(2x + 1) + 2x(2x + 1) + 4x + 2 + x - 1
= (2x^2 + 2x + 2)(2x + 1) + x - 1.
So the quotient is 2x^2 + 2x + 2, and the remainder is x - 1.

_____I x² + 2x + 5/2
2x +1 I 2x³ + 5x² + 7x + 1
_____I 2x³ + x²
_____I_____4x² + 7x + 1
_____I_____4x² + 2x
_____I__________5x + 1
_____I__________5x + 5/2
_____I_____________-3/2

x² + 2x + 5/2 - [ 3/2 ] / 2x + 1