Factor polynomial $$ -10x^{2}-34x-12 $$

$$ -10x^{2}-34x-12 = ( -2 )( 5x+2 )( x+3 ) $$

Step 1 :

After factoring out $ -2 $ we have:

$$ -10x^{2}-34x-12 = -2 ( 5x^{2}+17x+6 ) $$

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:

$$ a = 5 , b = 17 ~ \text{ and } ~ c = 6 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = 6} $.

$$ a \cdot c = 30 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 30 $ and add to $ b = 17 $.

Step 5: All pairs of numbers with a product of $ 30 $ are:

PRODUCT = 30
1     30	-1     -30
2     15	-2     -15
3     10	-3     -10
5     6	-5     -6

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 17 }$

PRODUCT = 30 and SUM = 17
1     30	-1     -30
2     15	-2     -15
3     10	-3     -10
5     6	-5     -6

Step 7: Replace middle term $ 17 x $ with $ 15x+2x $:

$$ 5x^{2}+17x+6 = 5x^{2}+15x+2x+6 $$

Step 8: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ 2 $ out of the last two terms.

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

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