Factor polynomial $$ 3x^{2}-5x-12 $$

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

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:

$$ a = 3 , b = -5 ~ \text{ and } ~ c = -12 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = -12} $.

$$ a \cdot c = -36 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -36 $ and add to $ b = -5 $.

Step 4: All pairs of numbers with a product of $ -36 $ are:

PRODUCT = -36
-1     36	1     -36
-2     18	2     -18
-3     12	3     -12
-4     9	4     -9
-6     6	6     -6

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

PRODUCT = -36 and SUM = -5
-1     36	1     -36
-2     18	2     -18
-3     12	3     -12
-4     9	4     -9
-6     6	6     -6

Step 6: Replace middle term $ -5 x $ with $ 4x-9x $:

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

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

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

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