Factor polynomial $$ 7x^{2}-64x-60 $$

$$ 7x^{2}-64x-60 = ( x-10 )( 7x+6 ) $$

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:

$$ a = 7 , b = -64 ~ \text{ and } ~ c = -60 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = -60} $.

$$ a \cdot c = -420 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -420 $ and add to $ b = -64 $.

Step 4: All pairs of numbers with a product of $ -420 $ are:

PRODUCT = -420
-1     420	1     -420
-2     210	2     -210
-3     140	3     -140
-4     105	4     -105
-5     84	5     -84
-6     70	6     -70
-7     60	7     -60
-10     42	10     -42
-12     35	12     -35
-14     30	14     -30
-15     28	15     -28
-20     21	20     -21

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -64 }$

PRODUCT = -420 and SUM = -64
-1     420	1     -420
-2     210	2     -210
-3     140	3     -140
-4     105	4     -105
-5     84	5     -84
-6     70	6     -70
-7     60	7     -60
-10     42	10     -42
-12     35	12     -35
-14     30	14     -30
-15     28	15     -28
-20     21	20     -21

Step 6: Replace middle term $ -64 x $ with $ 6x-70x $:

$$ 7x^{2}-64x-60 = 7x^{2}+6x-70x-60 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -10 $ out of the last two terms.

$$ 7x^{2}+6x-70x-60 = x\left(7x+6\right) -10\left(7x+6\right) = \left(x-10\right) \left(7x+6\right) $$

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