Factor polynomial $$ 30x^{3}-22x^{2}-28x $$

$$ 30x^{3}-22x^{2}-28x = 2x( 5x-7 )( 3x+2 ) $$

Step 1 :

After factoring out $ 2x $ we have:

$$ 30x^{3}-22x^{2}-28x = 2x ( 15x^{2}-11x-14 ) $$

Step 2 :

Step 2: 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:

$$ a = 15 , b = -11 ~ \text{ and } ~ c = -14 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 15 }$ by the constant term $\color{blue}{c = -14} $.

$$ a \cdot c = -210 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -210 $ and add to $ b = -11 $.

Step 5: All pairs of numbers with a product of $ -210 $ are:

PRODUCT = -210
-1     210	1     -210
-2     105	2     -105
-3     70	3     -70
-5     42	5     -42
-6     35	6     -35
-7     30	7     -30
-10     21	10     -21
-14     15	14     -15

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -11 }$

PRODUCT = -210 and SUM = -11
-1     210	1     -210
-2     105	2     -105
-3     70	3     -70
-5     42	5     -42
-6     35	6     -35
-7     30	7     -30
-10     21	10     -21
-14     15	14     -15

Step 7: Replace middle term $ -11 x $ with $ 10x-21x $:

$$ 15x^{2}-11x-14 = 15x^{2}+10x-21x-14 $$

Step 8: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -7 $ out of the last two terms.

$$ 15x^{2}+10x-21x-14 = 5x\left(3x+2\right) -7\left(3x+2\right) = \left(5x-7\right) \left(3x+2\right) $$

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