Verified answer

. . . . . 3x^2 + 4x + 2 <===== the second

. . . . . ___________________

x + 2 I 3x^3 + 10x^2 + 10x + 4

. . . . .-

. . . . . .3x^3 + 6x^2

. . . . . ____________________

. . . . . . .0 + 4x^2 + 10x + 4

. . . . . -

. . . . . . . . .4x^2 + 8x

. . . . . ____________________

. . . . . .. . . .0 + 2x + 4

. . . . . -

. . . . . . . .. . 0 + 2x + 4

. . . . . ____________________

. . . . . .. . . .. . 0 + 0

You can use synthetic division or factor. I will show you how to factor since synthetic division is difficult to show without visuals (the people above me show it pretty well but you can also follow this link to learn: http://www.wtamu.edu/academic/anns/mps/math/mathla...

3x^3 + 10x^2 + 10x + 4

1) Rewrite this as 3x^3 + 6x^2 + 4x^2 + 10x + 4

2) Break this down further to 3x^3 + 6x^2 + 4x^2 + 8x + 2x + 4

3) Group this as follows:

(3x^3 + 6x^2) + (4x^2 + 8x) + (2x + 4)

4) Within each of the groups, factor.

3x^2 (x + 2) + 4x(x + 2) + 2(x + 2)

5) Factor out the (x + 2) to get

(x + 2) (3x^2 + 4x + 2)

6) The division becomes:

(x + 2) (3x^2 + 4x + 2) / (x + 2)

7) Cancel out the (x +2) from the numerator and denominator.

3x^2 + 4x + 2 is your answer

Divide the polynomials by the comparing coefficients method:

(3x³ + 10x² + 10x + 4) / (x + 2) = Ax² + Bx + C

3x³ + 10x² + 10x + 4 = Ax²(x + 2) + Bx(x + 2) + C(x + 2)

3x³ + 10x² + 10x + 4 = Ax³ + 2Ax² + Bx² + 2Bx + Cx + 2C

3x³ + 10x² + 10x + 4 = Ax³ + (2A + B)x² + (2B + C)x + 2C

A = 3

2A + B = 10

B = 10 - 2A

B = 10 - 6

B = 4

2C = 4

C = 2

(3x³ + 10x² + 10x + 4) / (x + 2) = 3x² + 4x + 2

x+2 3x^3+10x^2+10x+4 3x^2+4x+2

- 3x^3+6x^2

--------------------

+4x^2+10x

- 4x^2+8x

-----------------

2x+4

-2x+4

--------------------

........ 3x^2 + 4x + 2

........____________________

x + 2| 3x^3 + 10x^2 + 10x + 4

.........3x^3 + 6x^2

......................4x^2 + 10x

......................4x^2 + 8x

...................................2x + 4

...................................2x + 4

...........................................0

Thus, (3x^3 + 10x^2 + 10x + 4)/(x + 2) = 3x^2 + 4x + 2

:)>