Factor polynomial $$ 9x^{2}-2x-32 $$

$$ 9x^{2}-2x-32 = ( x-2 )( 9x+16 ) $$

Step 1: 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:

$$ a = 9 , b = -2 ~ \text{ and } ~ c = -32 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 9 }$ by the constant term $\color{blue}{c = -32} $.

$$ a \cdot c = -288 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -288 $ and add to $ b = -2 $.

Step 4: All pairs of numbers with a product of $ -288 $ are:

PRODUCT = -288
-1     288	1     -288
-2     144	2     -144
-3     96	3     -96
-4     72	4     -72
-6     48	6     -48
-8     36	8     -36
-9     32	9     -32
-12     24	12     -24
-16     18	16     -18

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -2 }$

PRODUCT = -288 and SUM = -2
-1     288	1     -288
-2     144	2     -144
-3     96	3     -96
-4     72	4     -72
-6     48	6     -48
-8     36	8     -36
-9     32	9     -32
-12     24	12     -24
-16     18	16     -18

Step 6: Replace middle term $ -2 x $ with $ 16x-18x $:

$$ 9x^{2}-2x-32 = 9x^{2}+16x-18x-32 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -2 $ out of the last two terms.

$$ 9x^{2}+16x-18x-32 = x\left(9x+16\right) -2\left(9x+16\right) = \left(x-2\right) \left(9x+16\right) $$

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