# How do you solve x^3+3x^2<4 using a sign chart?

Jan 19, 2017

The answer is x in ] -oo,-2[uu] -2,1 [

#### Explanation:

Let's rewrite the inequality

${x}^{3} + 3 {x}^{2} - 4 < 0$

We must factorise the LHS

Let $f \left(x\right) = {x}^{3} + 3 {x}^{2} - 4$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R}$

$f \left(1\right) = 1 + 3 - 4 = 0$

So, $\left(x - 1\right)$ is a factor

To find the other factors, we do a long division

$\textcolor{w h i t e}{a a a a}$${x}^{3} + 3 {x}^{2}$$\textcolor{w h i t e}{a a a a}$$- 4$$\textcolor{w h i t e}{a a a a}$∣$x - 1$

$\textcolor{w h i t e}{a a a a}$${x}^{3} - {x}^{2}$$\textcolor{w h i t e}{a a a a}$color(white)(aaaaaaaa)∣${x}^{2} + 4 x + 4$

$\textcolor{w h i t e}{a a a a}$$0 + 4 {x}^{2}$$\textcolor{w h i t e}{a a a a}$color(white)(aaaaaaaa)

$\textcolor{w h i t e}{a a a a a a}$$+ 4 {x}^{2} - 4 x$

$\textcolor{w h i t e}{a a a a a a a a}$$+ 0 + 4 x - 4$

$\textcolor{w h i t e}{a a a a a a a a a a a a}$$+ 4 x - 4$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$+ 0 - 0$

Therefore,

${x}^{3} + 3 {x}^{2} - 4 = \left(x - 1\right) \left({x}^{2} + 4 x + 4\right) = \left(x - 1\right) {\left(x + 2\right)}^{2}$

So, we can make the sign chart

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

$\textcolor{w h i t e}{a a a a}$${\left(x + 2\right)}^{2}$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 1$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

Therefore,

$f \left(x\right) < 0$ when x in ] -oo,-2[uu] -2,1 [