Factor polynomial $$ 6x^{3}+7x^{2}+5x $$

$$ 6x^{3}+7x^{2}+5x = x( 6x^{2}+7x+5 ) $$

Step 1 :

After factoring out $ x $ we have:

$$ 6x^{3}+7x^{2}+5x = x ( 6x^{2}+7x+5 ) $$

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:

$$ a = 6 , b = 7 ~ \text{ and } ~ c = 5 $$

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

$$ a \cdot c = 30 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 30 $ and add to $ b = 7 $.

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 $\color{blue}{ b = 7 }$

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

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