Factor polynomial $$ 15x^{2}+29x+12 $$

$$ 15x^{2}+29x+12 = ( 5x+3 )( 3x+4 ) $$

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:

$$ a = 15 , b = 29 ~ \text{ and } ~ c = 12 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 15 }$ by the constant term $\color{blue}{c = 12} $.

$$ a \cdot c = 180 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 180 $ and add to $ b = 29 $.

Step 4: All pairs of numbers with a product of $ 180 $ are:

PRODUCT = 180
1     180	-1     -180
2     90	-2     -90
3     60	-3     -60
4     45	-4     -45
5     36	-5     -36
6     30	-6     -30
9     20	-9     -20
10     18	-10     -18
12     15	-12     -15

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 29 }$

PRODUCT = 180 and SUM = 29
1     180	-1     -180
2     90	-2     -90
3     60	-3     -60
4     45	-4     -45
5     36	-5     -36
6     30	-6     -30
9     20	-9     -20
10     18	-10     -18
12     15	-12     -15

Step 6: Replace middle term $ 29 x $ with $ 20x+9x $:

$$ 15x^{2}+29x+12 = 15x^{2}+20x+9x+12 $$

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

$$ 15x^{2}+20x+9x+12 = 5x\left(3x+4\right) + 3\left(3x+4\right) = \left(5x+3\right) \left(3x+4\right) $$

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