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

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

Step 1 :

After factoring out $ -1 $ we have:

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

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:

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

Step 3: Multiply the leading coefficient $\color{blue}{ a = 15 }$ by the constant term $\color{blue}{c = -6} $.

$$ a \cdot c = -90 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -90 $ and add to $ b = 1 $.

Step 5: 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 6: 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 7: Replace middle term $ 1 x $ with $ 10x-9x $:

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

Step 8: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -3 $ out of the last two terms.

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

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