Factor polynomial $$ 5x^{2}-32x-21 $$

$$ 5x^{2}-32x-21 = ( x-7 )( 5x+3 ) $$

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:

$$ a = 5 , b = -32 ~ \text{ and } ~ c = -21 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = -21} $.

$$ a \cdot c = -105 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -105 $ and add to $ b = -32 $.

Step 4: All pairs of numbers with a product of $ -105 $ are:

PRODUCT = -105
-1     105	1     -105
-3     35	3     -35
-5     21	5     -21
-7     15	7     -15

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -32 }$

PRODUCT = -105 and SUM = -32
-1     105	1     -105
-3     35	3     -35
-5     21	5     -21
-7     15	7     -15

Step 6: Replace middle term $ -32 x $ with $ 3x-35x $:

$$ 5x^{2}-32x-21 = 5x^{2}+3x-35x-21 $$

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

$$ 5x^{2}+3x-35x-21 = x\left(5x+3\right) -7\left(5x+3\right) = \left(x-7\right) \left(5x+3\right) $$

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