Question

factor the expression completely. use the graph to determine the roots for synthetic division. all factors use only integer coefficients and constants.
factor common factor
$-12x^4 + 54x^3 + 15x^2 + 75x$

Explanation:

Step1: Identify greatest common factor

The coefficients are -12, 54, 15, 75; their GCF is 3. All terms have an $x$, so GCF is $-3x$ (to make leading coefficient positive).

Step2: Factor out the GCF

$
-12x^4 + 54x^3 + 15x^2 + 75x = -3x(4x^3 - 18x^2 - 5x - 25)
$

Step3: Factor cubic via rational root test

Test root $x=5$: $4(5)^3 -18(5)^2 -5(5)-25=500-450-25-25=0$. Use synthetic division on $4x^3 -18x^2 -5x -25$ with root 5:

$$\begin{array}{r|rrrr} 5 & 4 & -18 & -5 & -25 \\ & & 20 & 10 & 25 \\ \hline & 4 & 2 & 5 & 0 \end{array}$$

Resulting quadratic: $4x^2+2x+5$ (no real integer roots).

Answer:

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