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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90x⁴ + 384x³ + 432x² - 96
synthetic division
6(15x⁴ + 64x³ + 72x² - 16)
factor out a root: -2
use the optional scratchpad.
enter the result:
6(□)

Explanation:

Step1: Factor out GCF first

$90x^4 + 384x^3 + 432x^2 - 96 = 6(15x^4 + 64x^3 + 72x^2 - 16)$

Step2: Synthetic division with root -2

Set up synthetic division for $15x^4 + 64x^3 + 72x^2 + 0x - 16$ with root $x=-2$:

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

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

Step3: Check for another root (x=2/3)

Test $x=\frac{2}{3}$ on $15x^3 + 34x^2 + 4x - 8$:
$15(\frac{2}{3})^3 + 34(\frac{2}{3})^2 + 4(\frac{2}{3}) - 8 = 15(\frac{8}{27}) + 34(\frac{4}{9}) + \frac{8}{3} - 8 = \frac{120}{27} + \frac{136}{9} + \frac{8}{3} - 8 = 0$
Synthetic division with $x=\frac{2}{3}$:

$$\begin{array}{r|rrrr} \frac{2}{3} & 15 & 34 & 4 & -8 \\ & & 10 & 29.33 & 22.89 \\ \hline & 15 & 44 & 33.33 & 14.89 \quad \text{(corrected: integer root test, use x=-2 again)} \end{array}$$

Re-test synthetic division on $15x^3 + 34x^2 + 4x - 8$ with $x=-2$:

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

Resulting polynomial: $15x^2 + 4x - 4$

Step4: Factor quadratic

$15x^2 + 4x - 4 = (3x + 2)(5x - 2)$

Step5: Combine all factors

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

Answer:

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

(Note: The question asks for the expression inside the 6( ), so the fully factored interior is $(x+2)^2(3x+2)(5x-2)$)