# How do you divide (-x^4-3x^3-2x^2-4x-7)/(x^2+3) ?

Nov 3, 2017

The result is $\left(- {x}^{2} - 3 x + 1\right)$ remainder $5 x - 10$.

#### Explanation:

Looking at the orders of the numerator and the denominator, we can see the result should be of order 2.
We can deduce the first term will be $- {x}^{2}$ in order for the first term of the product to be $- {x}^{4}$.
Looking at the constants in the numerator it seems there must be a remainder, since 3 is not a factor of 7.
We write the numerator as a product of two quadratics (the known denominator and the unknown quotient) plus a linear term (the unknown remainder):
$\left(- {x}^{4} - 3 {x}^{3} - 2 {x}^{2} - 4 x - 7\right)$
$= \left({x}^{2} + 3\right) \left(- {x}^{2} + A x + B\right) + C x + D$
Comparing coefficients of ${x}^{3}$, we see that: $A = - 3$
Comparing coefficients of ${x}^{2}$, we see that: $B - 3 = - 2 \Rightarrow B = 1$
Comparing coefficients of $x$ ,we see that: $3 A + C = - 4 \Rightarrow C = 5$
Comparing constants, we see that: $3 B + D = - 7 \Rightarrow D = - 10$
Hence the result is $\left(- {x}^{2} - 3 x + 1\right)$ remainder $5 x - 10$.

Nov 3, 2017

$- {x}^{2} - 3 x + 1 + \frac{5 x - 10}{{x}^{2} + 3}$

#### Explanation:

$\frac{- {x}^{4} - 3 {x}^{3} - 2 {x}^{2} - 4 x - 7}{\textcolor{g r e e n}{{x}^{2} + 3}}$

Note that I use place keepers such as $0 {x}^{3}$ for eas of alignment.

$\textcolor{w h i t e}{\text{dddddddddddddd}} - {x}^{4} - 3 {x}^{3} - 2 {x}^{2} - 4 x - 7$
$\textcolor{m a \ge n t a}{- {x}^{2}} \textcolor{g r e e n}{\left({x}^{2} + 3\right)} \to \textcolor{w h i t e}{\text{d") ul(-x^4+0x^3-3x^2larr" Subtract}}$
 color(white)("dddddddddddddddd")0-3x^3 +color(white)("d")x^2-4x-7
$\textcolor{m a \ge n t a}{- 3 x} \textcolor{g r e e n}{\left({x}^{2} + 3\right)} \to \textcolor{w h i t e}{\text{ddd.d") ul(-3x^3+0x^2-9xlarr" Subtract}}$
$\textcolor{w h i t e}{\text{dddddddddddddddddddd")0color(white)("d")+color(white)("d}} {x}^{2} + 5 x - 7$
$\textcolor{w h i t e}{1} \textcolor{m a \ge n t a}{+ 1} \textcolor{g r e e n}{\left({x}^{2} + 3\right)} \to \textcolor{w h i t e}{\text{ddddddddddd.d")ul( x^2+0x+3larr" Subtract}}$
color(white)("dddddddddddddddddddddddddd")0+color(magenta)(5x-10)larr" Stop"

Putting all this together we have:

color(magenta)(-x^2-3x+1+(5x-10)/(color(black)(color(green)(x^2+3)))