Step 1 :

After factoring out $ 2 $ we have:

$$ 12x^{2}+60x+70 = 2 ( 6x^{2}+30x+35 ) $$

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:

$$ a = 6 , b = 30 ~ \text{ and } ~ c = 35 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = 35} $.

$$ a \cdot c = 210 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 210 $ and add to $ b = 30 $.

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 $\color{blue}{ b = 30 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ 30 }$, we conclude the polynomial cannot be factored.

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