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:

$$ a = 6 , b = 23 ~ \text{ and } ~ c = 21 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = 21} $.

$$ a \cdot c = 126 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 126 $ and add to $ b = 23 $.

Step 4: All pairs of numbers with a product of $ 126 $ are:

PRODUCT = 126
1     126	-1     -126
2     63	-2     -63
3     42	-3     -42
6     21	-6     -21
7     18	-7     -18
9     14	-9     -14

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 23 }$

PRODUCT = 126 and SUM = 23
1     126	-1     -126
2     63	-2     -63
3     42	-3     -42
6     21	-6     -21
7     18	-7     -18
9     14	-9     -14

Step 6: Replace middle term $ 23 x $ with $ 14x+9x $:

$$ 6x^{2}+23x+21 = 6x^{2}+14x+9x+21 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ 3 $ out of the last two terms.

$$ 6x^{2}+14x+9x+21 = 2x\left(3x+7\right) + 3\left(3x+7\right) = \left(2x+3\right) \left(3x+7\right) $$

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