Factor polynomial $$ 3x^{2}+4x-7 $$

$$ 3x^{2}+4x-7 = ( x-1 )( 3x+7 ) $$

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:

$$ a = 3 , b = 4 ~ \text{ and } ~ c = -7 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = -7} $.

$$ a \cdot c = -21 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -21 $ and add to $ b = 4 $.

Step 4: All pairs of numbers with a product of $ -21 $ are:

PRODUCT = -21
-1     21	1     -21
-3     7	3     -7

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 4 }$

PRODUCT = -21 and SUM = 4
-1     21	1     -21
-3     7	3     -7

Step 6: Replace middle term $ 4 x $ with $ 7x-3x $:

$$ 3x^{2}+4x-7 = 3x^{2}+7x-3x-7 $$

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

$$ 3x^{2}+7x-3x-7 = x\left(3x+7\right) -1\left(3x+7\right) = \left(x-1\right) \left(3x+7\right) $$

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