Factor polynomial $$ 5n^{2}+19n+12 $$

$$ 5n^{2}+19n+12 = ( 5n+4 )( n+3 ) $$

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:

$$ a = 5 , b = 19 ~ \text{ and } ~ c = 12 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = 12} $.

$$ a \cdot c = 60 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 60 $ and add to $ b = 19 $.

Step 4: All pairs of numbers with a product of $ 60 $ are:

PRODUCT = 60
1     60	-1     -60
2     30	-2     -30
3     20	-3     -20
4     15	-4     -15
5     12	-5     -12
6     10	-6     -10

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 19 }$

PRODUCT = 60 and SUM = 19
1     60	-1     -60
2     30	-2     -30
3     20	-3     -20
4     15	-4     -15
5     12	-5     -12
6     10	-6     -10

Step 6: Replace middle term $ 19 x $ with $ 15x+4x $:

$$ 5x^{2}+19x+12 = 5x^{2}+15x+4x+12 $$

Step 7: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ 4 $ out of the last two terms.

$$ 5x^{2}+15x+4x+12 = 5x\left(x+3\right) + 4\left(x+3\right) = \left(5x+4\right) \left(x+3\right) $$

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