Start with dividing the term with the highest exponent by the term with the highest exponent, so here, divide 3x^4 by x.
3x^4 -: x = 3x^3
Multiply 3x^3 with x+2: 3x^3(x+2) = 3x^4 + 6 x^3.
Your x^3 term is 2x^3, so to compute your remainder for the x^3 term, compute 2x^3 - 6x^3 = - 4 x^3.
So, you can already transform your numerator as follows:
3x^4 + 2x^3 - x^2 + 2x - 9 = color(green)(3x^3(x+2)) color(red)( - 4x^3 - x^2 + 2x - 9).
The green coloured terms are already done, now take care of the red coloured terms.
Again, divide the term with the highest exponent:
(-4x^3) -: x = -4 x^2
Multiply with x+2:
-4 x^2 (x+2) = - 4 x^3 - 8 x^2
The remainder for your x^2 term is:
- x^2 - (-8 x ^2) = 7 x^2.
So,
3x^4 + 2x^3 - x^2 + 2x - 9
= color(green)(3x^3(x+2)) - 4x^3 - x^2 + 2x - 9
= color(green)(3x^3(x+2) - 4x^2(x+2)) color(red)( + 7x^2 + 2x - 9).
Repeate the procedure with the remaining red term.
Divide: 7x^2 -: x = 7x
Multiply: 7x (x+2) = 7x^2 + 14x
Remainder: 2x - 14x =- 12x
In total:
3x^4 + 2x^3 - x^2 + 2x - 9
= color(green)(3x^3(x+2)) - 4x^3 - x^2 + 2x - 9
= color(green)(3x^3(x+2) - 4x^2(x+2)) + 7x^2 + 2x - 9
= color(green)(3x^3(x+2) - 4x^2(x+2) + 7x( x + 2)) color(red)(- 12x - 9).
Last step: try to divide -12x - 9 by x + 2:
Divide: -12x -: x = -12
Multiply: -12(x+2) = -12 x - 24
Remainder: -9 - (-24) = 15
As the degree of the remainder (0, since no x or rather x^0) is lower than the degree of the denominator (degree 1, highest exponent x^1), you can stop here.
The result is:
3x^4 + 2x^3 - x^2 + 2x - 9
= color(green)(3x^3(x+2)) - 4x^3 - x^2 + 2x - 9
= color(green)(3x^3(x+2) - 4x^2(x+2)) + 7x^2 + 2x - 9
= color(green)(3x^3(x+2) - 4x^2(x+2) + 7x( x + 2)) - 12x - 9
= color(green)(3x^3(x+2) - 4x^2(x+2) + 7x( x + 2) - 12(x + 2)) + 15
So, the result of
(3x^4 + 2x^3 - x^2 + 2x - 9) -: (x+2) is
3x^3 - 4x^2 + 7x - 12 with the remainder 15, or:
(3x^4 + 2x^3 - x^2 + 2x - 9) / (x+2) = 3x^3 - 4x^2 + 7x - 12 + 15/(x+2)
Hope that this helped!
