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:

$$ a = 12 , b = 41 ~ \text{ and } ~ c = 35 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 12 }$ by the constant term $\color{blue}{c = 35} $.

$$ a \cdot c = 420 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 420 $ and add to $ b = 41 $.

Step 4: 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 5: Find out which factor pair sums up to $\color{blue}{ b = 41 }$

PRODUCT = 420 and SUM = 41
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: Replace middle term $ 41 x $ with $ 21x+20x $:

$$ 12x^{2}+41x+35 = 12x^{2}+21x+20x+35 $$

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ 5 $ out of the last two terms.

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

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