Factor polynomial $$ 6x^{2}-x-15 $$

$$ 6x^{2}-x-15 = ( 3x-5 )( 2x+3 ) $$

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:

$$ a = 6 , b = -1 ~ \text{ and } ~ c = -15 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = -15} $.

$$ a \cdot c = -90 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -90 $ and add to $ b = -1 $.

Step 4: All pairs of numbers with a product of $ -90 $ are:

PRODUCT = -90
-1     90	1     -90
-2     45	2     -45
-3     30	3     -30
-5     18	5     -18
-6     15	6     -15
-9     10	9     -10

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -1 }$

PRODUCT = -90 and SUM = -1
-1     90	1     -90
-2     45	2     -45
-3     30	3     -30
-5     18	5     -18
-6     15	6     -15
-9     10	9     -10

Step 6: Replace middle term $ -1 x $ with $ 9x-10x $:

$$ 6x^{2}-x-15 = 6x^{2}+9x-10x-15 $$

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ -5 $ out of the last two terms.

$$ 6x^{2}+9x-10x-15 = 3x\left(2x+3\right) -5\left(2x+3\right) = \left(3x-5\right) \left(2x+3\right) $$

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