Factor polynomial $$ 18v^{9}+33v^{8}+14v^{7} $$

$$ 18v^{9}+33v^{8}+14v^{7} = v^{7}( 3v+2 )( 6v+7 ) $$

Step 1 :

After factoring out $ v^{7} $ we have:

$$ 18v^{9}+33v^{8}+14v^{7} = v^{7} ( 18v^{2}+33v+14 ) $$

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:

$$ a = 18 , b = 33 ~ \text{ and } ~ c = 14 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 18 }$ by the constant term $\color{blue}{c = 14} $.

$$ a \cdot c = 252 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 252 $ and add to $ b = 33 $.

Step 5: All pairs of numbers with a product of $ 252 $ are:

PRODUCT = 252
1     252	-1     -252
2     126	-2     -126
3     84	-3     -84
4     63	-4     -63
6     42	-6     -42
7     36	-7     -36
9     28	-9     -28
12     21	-12     -21
14     18	-14     -18

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 33 }$

PRODUCT = 252 and SUM = 33
1     252	-1     -252
2     126	-2     -126
3     84	-3     -84
4     63	-4     -63
6     42	-6     -42
7     36	-7     -36
9     28	-9     -28
12     21	-12     -21
14     18	-14     -18

Step 7: Replace middle term $ 33 x $ with $ 21x+12x $:

$$ 18x^{2}+33x+14 = 18x^{2}+21x+12x+14 $$

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

$$ 18x^{2}+21x+12x+14 = 3x\left(6x+7\right) + 2\left(6x+7\right) = \left(3x+2\right) \left(6x+7\right) $$

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