Factor polynomial $$ 12x^{2}-9x+15 $$

$$ 12x^{2}-9x+15 = 3( 4x^{2}-3x+5 ) $$

Step 1 :

After factoring out $ 3 $ we have:

$$ 12x^{2}-9x+15 = 3 ( 4x^{2}-3x+5 ) $$

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:

$$ a = 4 , b = -3 ~ \text{ and } ~ c = 5 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 5} $.

$$ a \cdot c = 20 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 20 $ and add to $ b = -3 $.

Step 5: All pairs of numbers with a product of $ 20 $ are:

PRODUCT = 20
1     20	-1     -20
2     10	-2     -10
4     5	-4     -5

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -3 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -3 }$, we conclude the polynomial cannot be factored.

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