Factor polynomial $$ 42p^{2}+456p+360 $$

$$ 42p^{2}+456p+360 = 6( 7p+6 )( p+10 ) $$

Step 1 :

After factoring out $ 6 $ we have:

$$ 42p^{2}+456p+360 = 6 ( 7p^{2}+76p+60 ) $$

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:

$$ a = 7 , b = 76 ~ \text{ and } ~ c = 60 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = 60} $.

$$ a \cdot c = 420 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 420 $ and add to $ b = 76 $.

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 $\color{blue}{ b = 76 }$

PRODUCT = 420 and SUM = 76
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 $ 76 x $ with $ 70x+6x $:

$$ 7x^{2}+76x+60 = 7x^{2}+70x+6x+60 $$

Step 8: Apply factoring by grouping. Factor $ 7x $ out of the first two terms and $ 6 $ out of the last two terms.

$$ 7x^{2}+70x+6x+60 = 7x\left(x+10\right) + 6\left(x+10\right) = \left(7x+6\right) \left(x+10\right) $$

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