Factor polynomial $$ 18a^{2}-48a-32 $$

$$ 18a^{2}-48a-32 = 2( 9a^{2}-24a-16 ) $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 18a^{2}-48a-32 = 2 ( 9a^{2}-24a-16 ) $$

Step 2 :

Step 2: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 9 , b = -24 ~ \text{ and } ~ c = -16 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 9 }$ by the constant term $\color{blue}{c = -16} $.

$$ a \cdot c = -144 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -144 $ and add to $ b = -24 $.

Step 5: All pairs of numbers with a product of $ -144 $ are:

PRODUCT = -144
-1     144	1     -144
-2     72	2     -72
-3     48	3     -48
-4     36	4     -36
-6     24	6     -24
-8     18	8     -18
-9     16	9     -16
-12     12	12     -12

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -24 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -24 }$, we conclude the polynomial cannot be factored.

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