Factor polynomial $$ 3x^{2}+2x-1 $$

$$ 3x^{2}+2x-1 = ( 3x-1 )( x+1 ) $$

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:

$$ a = 3 , b = 2 ~ \text{ and } ~ c = -1 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = -1} $.

$$ a \cdot c = -3 $$

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

Step 4: All pairs of numbers with a product of $ -3 $ are:

PRODUCT = -3
-1     3	1     -3

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

PRODUCT = -3 and SUM = 2
-1     3	1     -3

Step 6: Replace middle term $ 2 x $ with $ 3x-x $:

$$ 3x^{2}+2x-1 = 3x^{2}+3x-x-1 $$

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ -1 $ out of the last two terms.

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

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