How do you divide (-x^4-3x^3-2x^2-2x+3)/(x^2+3) ?

Oct 17, 2017

$- {x}^{2} - 3 x + 1 + \frac{7 x}{{x}^{2} + 3}$

Explanation:

Given: (color(blue)(-x^4-3x^3-2x^2-2x+3))/(color(green)(x^2+3)

Note that I use place keepers that have no value. This is to assist with formatting. Example $0 {x}^{3}$

$\textcolor{w h i t e}{\text{dddddddddddddddd}} \textcolor{b l u e}{- {x}^{4} - 3 {x}^{3} - 2 {x}^{2} - 2 x + 3}$
$\textcolor{m a \ge n t a}{- \left({x}^{2}\right)} \textcolor{g r e e n}{\left({x}^{2} + 3\right)} \to \textcolor{w h i t e}{\text{.d")ul(-x^4+0x^3-3x^2 larr" Subtract}}$
$\textcolor{w h i t e}{\text{ddddddddddddddddddd") 0-3x^3+color(white)(".}} {x}^{2} - 2 x + 3$
$\textcolor{m a \ge n t a}{- 3 x} \textcolor{g r e e n}{\left({x}^{2} + 3\right)} \to \textcolor{w h i t e}{\text{ddddddd")-ul(3x^3+0x^2-9x larr" Subtract}}$
$\textcolor{w h i t e}{\text{dddddddddddddddddddddddd")0 +x^2color(white)("d}} + 7 x + 3$
$\textcolor{m a \ge n t a}{+ 1} \textcolor{g r e e n}{\left({x}^{2} + 3\right)} \to \textcolor{w h i t e}{\text{ddddddddddddddd")ul(x^2color(white)("d")+0x+3 larr" Sub.}}$
$\textcolor{w h i t e}{\text{ddddddddddd")color(magenta)("Remainder"->color(white)("dddddd")0color(white)("d}} \textcolor{m a \ge n t a}{+ 7 x + 0}$

color(magenta)(-x^2-3x+1+(7x)/(color(green)(x^2+3))