Question

factor the expression completely. use the graph to determine the roots for synthetic division. all factors use only integer coefficients and constants.
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factor common factor
$90x^4 + 384x^3 + 432x^2 - 96$
use the optional scratchpad.
enter the result.

Explanation:

Step1: Factor out GCF

Find the greatest common factor (GCF) of all terms: $\text{GCF}(90, 384, 432, 96) = 6$
$90x^4 + 384x^3 + 432x^2 - 96 = 6(15x^4 + 64x^3 + 72x^2 - 16)$

Step2: Identify rational root

Use Rational Root Theorem: possible roots are $\pm1, \pm2, \pm4, \pm8, \pm16, \pm\frac{1}{3}, \pm\frac{2}{3}, ...$
Test $x=-2$: $15(-2)^4 + 64(-2)^3 + 72(-2)^2 -16 = 15(16)+64(-8)+72(4)-16=240-512+288-16=0$
So $(x+2)$ is a factor.

Step3: Synthetic division (degree 4)

Divide $15x^4 + 64x^3 + 72x^2 -16$ by $(x+2)$:

$$\begin{array}{r|rrrrr} -2 & 15 & 64 & 72 & 0 & -16 \\ & & -30 & -68 & -8 & 16 \\ \hline & 15 & 34 & 4 & -8 & 0 \end{array}$$

Result: $15x^3 + 34x^2 + 4x - 8$

Step4: Factor cubic polynomial

Test $x=\frac{2}{3}$ for $15x^3 + 34x^2 + 4x - 8$:
$15(\frac{2}{3})^3 +34(\frac{2}{3})^2 +4(\frac{2}{3}) -8 = 15(\frac{8}{27})+34(\frac{4}{9})+\frac{8}{3}-8=\frac{120}{27}+\frac{136}{9}+\frac{8}{3}-8=\frac{40}{9}+\frac{136}{9}+\frac{24}{9}-\frac{72}{9}=\frac{128}{9}
eq0$
Test $x=\frac{1}{3}$: $15(\frac{1}{3})^3 +34(\frac{1}{3})^2 +4(\frac{1}{3}) -8=\frac{15}{27}+\frac{34}{9}+\frac{4}{3}-8=\frac{5}{9}+\frac{34}{9}+\frac{12}{9}-\frac{72}{9}=-\frac{21}{9}
eq0$
Test $x=-2$ again: $15(-2)^3 +34(-2)^2 +4(-2)-8=15(-8)+34(4)-8-8=-120+136-16=0$
So $(x+2)$ is a factor again.

Step5: Synthetic division (degree 3)

Divide $15x^3 + 34x^2 + 4x - 8$ by $(x+2)$:

$$\begin{array}{r|rrrr} -2 & 15 & 34 & 4 & -8 \\ & & -30 & -8 & 8 \\ \hline & 15 & 4 & -4 & 0 \end{array}$$

Result: $15x^2 + 4x - 4$

Step6: Factor quadratic polynomial

Factor $15x^2 + 4x - 4$:
Find two numbers: $10$ and $-6$ (since $10\times(-6)=-60$, $10+(-6)=4$)
$15x^2 +10x -6x -4 = 5x(3x+2)-2(3x+2)=(5x-2)(3x+2)$

Step7: Combine all factors

Put all factored terms together with the initial GCF:
$6(x+2)(x+2)(5x-2)(3x+2)=6(x+2)^2(5x-2)(3x+2)$

Answer:

$6(x+2)^2(3x+2)(5x-2)$