Factor polynomial $$ 7z^{2}-5z-2 $$

$$ 7z^{2}-5z-2 = ( z-1 )( 7z+2 ) $$

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:

$$ a = 7 , b = -5 ~ \text{ and } ~ c = -2 $$

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

$$ a \cdot c = -14 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -14 $ and add to $ b = -5 $.

Step 4: All pairs of numbers with a product of $ -14 $ are:

PRODUCT = -14
-1     14	1     -14
-2     7	2     -7

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

PRODUCT = -14 and SUM = -5
-1     14	1     -14
-2     7	2     -7

Step 6: Replace middle term $ -5 x $ with $ 2x-7x $:

$$ 7x^{2}-5x-2 = 7x^{2}+2x-7x-2 $$

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

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

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