# If the quotient of two polynomials is 4x^2-x-7+(11x+15)/(x^2+x+2), what are the two polynomials?

Apr 2, 2017

$\left(4 {x}^{4} + 3 {x}^{3} + 0 {x}^{2} + 2 x + 1\right) \div \left({x}^{2} + x + 2\right)$

#### Explanation:

Given:$\text{ } 4 {x}^{2} - x - 7 + \frac{11 x + 15}{{x}^{2} + x + 2}$

The $11 x + 15$ is the remainder so the denominator must be the divisor. Thus we need to build each term by using ${x}^{2} + x + 2$

Note that I use a place holder. For example: $0 {x}^{4}$

color(white)(.)

$4 {x}^{2} \left({x}^{2} + x + 2\right) \to 4 {x}^{4} + 4 {x}^{3} + 8 {x}^{2}$
$- x \left({x}^{2} + x + 2\right) \to 0 {x}^{4} - \textcolor{w h i t e}{4} {x}^{3} - \textcolor{w h i t e}{8} {x}^{2} - 2 x$
$- 7 \left({x}^{2} + x + 2\right) \to 0 {x}^{4} + 0 {x}^{3} - 7 {x}^{2} - 7 x - 14$
$\text{The remainder"->ul(color(white)(.)0x^4+0x^3+0x^2+11x+15) larr" Add}$
$\text{ } 4 {x}^{4} + 3 {x}^{3} + 0 {x}^{2} + 2 x + 1$