Factor polynomial $$ 24x^{2}+16x-78 $$

$$ 24x^{2}+16x-78 = 2( 2x-3 )( 6x+13 ) $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 24x^{2}+16x-78 = 2 ( 12x^{2}+8x-39 ) $$

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:

$$ a = 12 , b = 8 ~ \text{ and } ~ c = -39 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 12 }$ by the constant term $\color{blue}{c = -39} $.

$$ a \cdot c = -468 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -468 $ and add to $ b = 8 $.

Step 5: All pairs of numbers with a product of $ -468 $ are:

PRODUCT = -468
-1     468	1     -468
-2     234	2     -234
-3     156	3     -156
-4     117	4     -117
-6     78	6     -78
-9     52	9     -52
-12     39	12     -39
-13     36	13     -36
-18     26	18     -26

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 8 }$

PRODUCT = -468 and SUM = 8
-1     468	1     -468
-2     234	2     -234
-3     156	3     -156
-4     117	4     -117
-6     78	6     -78
-9     52	9     -52
-12     39	12     -39
-13     36	13     -36
-18     26	18     -26

Step 7: Replace middle term $ 8 x $ with $ 26x-18x $:

$$ 12x^{2}+8x-39 = 12x^{2}+26x-18x-39 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -3 $ out of the last two terms.

$$ 12x^{2}+26x-18x-39 = 2x\left(6x+13\right) -3\left(6x+13\right) = \left(2x-3\right) \left(6x+13\right) $$

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