Factor polynomial $$ 12x^{2}+69x+45 $$

$$ 12x^{2}+69x+45 = 3( 4x+3 )( x+5 ) $$

Step 1 :

After factoring out $ 3 $ we have:

$$ 12x^{2}+69x+45 = 3 ( 4x^{2}+23x+15 ) $$

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:

$$ a = 4 , b = 23 ~ \text{ and } ~ c = 15 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 15} $.

$$ a \cdot c = 60 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 60 $ and add to $ b = 23 $.

Step 5: 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 6: Find out which factor pair sums up to $\color{blue}{ b = 23 }$

PRODUCT = 60 and SUM = 23
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 7: Replace middle term $ 23 x $ with $ 20x+3x $:

$$ 4x^{2}+23x+15 = 4x^{2}+20x+3x+15 $$

Step 8: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ 3 $ out of the last two terms.

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

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