Factor polynomial $$ 15g^{2}-g-2 $$

$$ 15g^{2}-g-2 = ( 5g-2 )( 3g+1 ) $$

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:

$$ a = 15 , b = -1 ~ \text{ and } ~ c = -2 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 15 }$ by the constant term $\color{blue}{c = -2} $.

$$ a \cdot c = -30 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -30 $ and add to $ b = -1 $.

Step 4: All pairs of numbers with a product of $ -30 $ are:

PRODUCT = -30
-1     30	1     -30
-2     15	2     -15
-3     10	3     -10
-5     6	5     -6

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -1 }$

PRODUCT = -30 and SUM = -1
-1     30	1     -30
-2     15	2     -15
-3     10	3     -10
-5     6	5     -6

Step 6: Replace middle term $ -1 x $ with $ 5x-6x $:

$$ 15x^{2}-x-2 = 15x^{2}+5x-6x-2 $$

Step 7: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -2 $ out of the last two terms.

$$ 15x^{2}+5x-6x-2 = 5x\left(3x+1\right) -2\left(3x+1\right) = \left(5x-2\right) \left(3x+1\right) $$

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