Factor polynomial $$ 12b^{2}-60b-45 $$

$$ 12b^{2}-60b-45 = 3( 4b^{2}-20b-15 ) $$

Step 1 :

After factoring out $ 3 $ we have:

$$ 12b^{2}-60b-45 = 3 ( 4b^{2}-20b-15 ) $$

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:

$$ a = 4 , b = -20 ~ \text{ and } ~ c = -15 $$

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 = -20 $.

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 $\color{blue}{ b = -20 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -20 }$, we conclude the polynomial cannot be factored.

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