Factor polynomial $$ 24c^{2}+26c-15 $$

$$ 24c^{2}+26c-15 = ( 12c-5 )( 2c+3 ) $$

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:

$$ a = 24 , b = 26 ~ \text{ and } ~ c = -15 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 24 }$ by the constant term $\color{blue}{c = -15} $.

$$ a \cdot c = -360 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -360 $ and add to $ b = 26 $.

Step 4: All pairs of numbers with a product of $ -360 $ are:

PRODUCT = -360
-1     360	1     -360
-2     180	2     -180
-3     120	3     -120
-4     90	4     -90
-5     72	5     -72
-6     60	6     -60
-8     45	8     -45
-9     40	9     -40
-10     36	10     -36
-12     30	12     -30
-15     24	15     -24
-18     20	18     -20

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 26 }$

PRODUCT = -360 and SUM = 26
-1     360	1     -360
-2     180	2     -180
-3     120	3     -120
-4     90	4     -90
-5     72	5     -72
-6     60	6     -60
-8     45	8     -45
-9     40	9     -40
-10     36	10     -36
-12     30	12     -30
-15     24	15     -24
-18     20	18     -20

Step 6: Replace middle term $ 26 x $ with $ 36x-10x $:

$$ 24x^{2}+26x-15 = 24x^{2}+36x-10x-15 $$

Step 7: Apply factoring by grouping. Factor $ 12x $ out of the first two terms and $ -5 $ out of the last two terms.

$$ 24x^{2}+36x-10x-15 = 12x\left(2x+3\right) -5\left(2x+3\right) = \left(12x-5\right) \left(2x+3\right) $$

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