# How do you solve (n^3-2n^2-n+2)/(n^3+3n^2+4n+12)<0?

Nov 11, 2017

The solution is n in (-3,-1) uu1,2)

#### Explanation:

Factorise the numerator and the denominator

${n}^{3} - 2 {n}^{2} - n + 2 = {n}^{3} - n - 2 {n}^{2} + 2$

$= n \left({n}^{2} - 1\right) - 2 \left({n}^{2} - 1\right)$

$= \left({n}^{2} - 1\right) \left(n - 2\right)$

$= \left(n + 1\right) \left(n - 1\right) \left(n - 2\right)$

Proceed with the denominator

${n}^{3} + 3 {n}^{2} + + 4 n + 12 = {n}^{3} + 4 n + 3 {n}^{2} + 12$

$= n \left({n}^{2} + 4\right) + 3 \left({n}^{2} + 4\right)$

$= \left({n}^{2} + 4\right) \left(n + 3\right)$

Let $f \left(n\right) = \frac{{n}^{3} - 2 {n}^{2} - n + 2}{{n}^{3} + 3 {n}^{2} + + 4 n + 12} = \frac{\left(n + 1\right) \left(n - 1\right) \left(n - 2\right)}{\left({n}^{2} + 4\right) \left(n + 3\right)}$

Build a sign chart

$\textcolor{w h i t e}{a a a a}$$n$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a}$$- 3$$\textcolor{w h i t e}{a a a a}$$- 1$$\textcolor{w h i t e}{a a a a}$$1$$\textcolor{w h i t e}{a a a a a}$$2$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$n + 3$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$n + 1$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$n - 1$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$n - 2$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(n\right)$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$

Therefore,

$f \left(n\right) < 0$ when n in (-3,-1) uu1,2)

graph{(x^3-2x^2-x+2)/(x^3+4x+3x^2+12) [-12.66, 12.65, -6.33, 6.33]}