Factor polynomial $$ 24x^{3}+22x^{2}-10x $$

$$ 24x^{3}+22x^{2}-10x = 2x( 3x-1 )( 4x+5 ) $$

Step 1 :

After factoring out $ 2x $ we have:

$$ 24x^{3}+22x^{2}-10x = 2x ( 12x^{2}+11x-5 ) $$

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:

$$ a = 12 , b = 11 ~ \text{ and } ~ c = -5 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 12 }$ by the constant term $\color{blue}{c = -5} $.

$$ a \cdot c = -60 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -60 $ and add to $ b = 11 $.

Step 5: All pairs of numbers with a product of $ -60 $ are:

PRODUCT = -60
-1     60	1     -60
-2     30	2     -30
-3     20	3     -20
-4     15	4     -15
-5     12	5     -12
-6     10	6     -10

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

PRODUCT = -60 and SUM = 11
-1     60	1     -60
-2     30	2     -30
-3     20	3     -20
-4     15	4     -15
-5     12	5     -12
-6     10	6     -10

Step 7: Replace middle term $ 11 x $ with $ 15x-4x $:

$$ 12x^{2}+11x-5 = 12x^{2}+15x-4x-5 $$

Step 8: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ -1 $ out of the last two terms.

$$ 12x^{2}+15x-4x-5 = 3x\left(4x+5\right) -1\left(4x+5\right) = \left(3x-1\right) \left(4x+5\right) $$

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