Factor polynomial $$ 12b^{2}+5b-2 $$

$$ 12b^{2}+5b-2 = ( 4b-1 )( 3b+2 ) $$

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:

$$ a = 12 , b = 5 ~ \text{ and } ~ c = -2 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 12 }$ by the constant term $\color{blue}{c = -2} $.

$$ a \cdot c = -24 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -24 $ and add to $ b = 5 $.

Step 4: All pairs of numbers with a product of $ -24 $ are:

PRODUCT = -24
-1     24	1     -24
-2     12	2     -12
-3     8	3     -8
-4     6	4     -6

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

PRODUCT = -24 and SUM = 5
-1     24	1     -24
-2     12	2     -12
-3     8	3     -8
-4     6	4     -6

Step 6: Replace middle term $ 5 x $ with $ 8x-3x $:

$$ 12x^{2}+5x-2 = 12x^{2}+8x-3x-2 $$

Step 7: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ -1 $ out of the last two terms.

$$ 12x^{2}+8x-3x-2 = 4x\left(3x+2\right) -1\left(3x+2\right) = \left(4x-1\right) \left(3x+2\right) $$

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