Factor polynomial $$ 12x^{2}+35x+25 $$

$$ 12x^{2}+35x+25 = ( 4x+5 )( 3x+5 ) $$

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:

$$ a = 12 , b = 35 ~ \text{ and } ~ c = 25 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 12 }$ by the constant term $\color{blue}{c = 25} $.

$$ a \cdot c = 300 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 300 $ and add to $ b = 35 $.

Step 4: All pairs of numbers with a product of $ 300 $ are:

PRODUCT = 300
1     300	-1     -300
2     150	-2     -150
3     100	-3     -100
4     75	-4     -75
5     60	-5     -60
6     50	-6     -50
10     30	-10     -30
12     25	-12     -25
15     20	-15     -20

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 35 }$

PRODUCT = 300 and SUM = 35
1     300	-1     -300
2     150	-2     -150
3     100	-3     -100
4     75	-4     -75
5     60	-5     -60
6     50	-6     -50
10     30	-10     -30
12     25	-12     -25
15     20	-15     -20

Step 6: Replace middle term $ 35 x $ with $ 20x+15x $:

$$ 12x^{2}+35x+25 = 12x^{2}+20x+15x+25 $$

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

$$ 12x^{2}+20x+15x+25 = 4x\left(3x+5\right) + 5\left(3x+5\right) = \left(4x+5\right) \left(3x+5\right) $$

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