Question

Solve for x.x³ + x² = 4x + 4x = [?], [ ], [ ]Order your answers from smallest tolargest. Enter the number that belongs inthe green box.Note: Enter any fractional answers in decimal form.Enter

Answer 1

To solve this problem we will factor the given polynomial.

First, we rewrite the given equation as follows:

`x^3+x^2-4x-4=0.`

Now, notice that:

`x^3+x^2-4x-4=(x+1)(x+2)(x-2).`

Therefore:

`\begin{gathered} x^3+x^2-4x-4=0\text{ if and only if} \\ \left(x+1\right)\left(x+2\right)\left(x-2\right)=0. \end{gathered}`

Recall that:

`ab=0,\text{ if and only if a=0 or b=0.}`

Therefore:

`\left(x+1\right)\left(x+2\right)\left(x-2\right)=0\text{ if \lparen x+1\rparen=0 or \lparen x+2\rparen=0 or \lparen x-2\rparen=0.}`

Solving each equation for x, we get:

`x=-1,\text{ x=-2, x=2.}`

Answer:

`?=-2.`