# How do you divide \frac { x ^ { 2} + 3x + 2} { 3x ^ { 2} + x - 2} \div ( x + 2)?

Apr 14, 2018

$\left(3 x - 2\right)$

#### Explanation:

When you are simplifying algebraic fractions, factorise first wherever possible.

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

$\frac{\left(x + 2\right) \left(x + 1\right)}{\left(3 x - 2\right) \left(x + 1\right)} \times \frac{1}{\left(x + 2\right)} \text{ } \leftarrow$ cancel common factors

$= \frac{\left(x + 2\right) \left(x + 1\right)}{\left(3 x - 2\right) \left(x + 1\right)} \times \frac{1}{\left(x + 2\right)}$

$= \frac{\cancel{\left(x + 2\right)} \cancel{\left(x + 1\right)}}{\left(3 x - 2\right) \cancel{\left(x + 1\right)}} \times \frac{1}{\cancel{\left(x + 2\right)}}$

$= \left(3 x - 2\right)$

Apr 14, 2018

Lets begin by factorizing each quadratic trinomial,

color(red)(x^2+3x+2

$\implies {x}^{2} + x + 2 x + 2$

$\implies x \left(x + 1\right) + 2 \left(x + 1\right)$

=> color(red)((x+ 2)(x+1)

color(magenta)(3x^2+x−2

=>3x^2+3x-2x−2

$\implies 3 x \left(x + 1\right) - 2 \left(x + 1\right)$

=> color(magenta)((3x-2)(x+1)

$\text{Now, according to the question, we need to find,}$

(color(red)(x^2+3x+2))/(color(magenta)(3x^2+x−2)) xx 1/(x+2)

=> color(red)(cancel((x+ 2))cancel((x+1)))/ color(magenta)((3x-2)cancel((x+1)))xx 1/(cancel((x+2))

=> 1/(3x-2