Step 1 :

After factoring out $ 2 $ we have:

$$ 24g^{2}-2g-70 = 2 ( 12g^{2}-g-35 ) $$

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:

$$ a = 12 , b = -1 ~ \text{ and } ~ c = -35 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 12 }$ by the constant term $\color{blue}{c = -35} $.

$$ a \cdot c = -420 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -420 $ and add to $ b = -1 $.

Step 5: All pairs of numbers with a product of $ -420 $ are:

PRODUCT = -420
-1     420	1     -420
-2     210	2     -210
-3     140	3     -140
-4     105	4     -105
-5     84	5     -84
-6     70	6     -70
-7     60	7     -60
-10     42	10     -42
-12     35	12     -35
-14     30	14     -30
-15     28	15     -28
-20     21	20     -21

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -1 }$

PRODUCT = -420 and SUM = -1
-1     420	1     -420
-2     210	2     -210
-3     140	3     -140
-4     105	4     -105
-5     84	5     -84
-6     70	6     -70
-7     60	7     -60
-10     42	10     -42
-12     35	12     -35
-14     30	14     -30
-15     28	15     -28
-20     21	20     -21

Step 7: Replace middle term $ -1 x $ with $ 20x-21x $:

$$ 12x^{2}-x-35 = 12x^{2}+20x-21x-35 $$

Step 8: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ -7 $ out of the last two terms.

$$ 12x^{2}+20x-21x-35 = 4x\left(3x+5\right) -7\left(3x+5\right) = \left(4x-7\right) \left(3x+5\right) $$

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