Factor polynomial $$ 42y^{5}+36y^{4}-60y^{3} $$

$$ 42y^{5}+36y^{4}-60y^{3} = 6y^{3}( 7y^{2}+6y-10 ) $$

Step 1 :

After factoring out $ 6y^{3} $ we have:

$$ 42y^{5}+36y^{4}-60y^{3} = 6y^{3} ( 7y^{2}+6y-10 ) $$

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:

$$ a = 7 , b = 6 ~ \text{ and } ~ c = -10 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = -10} $.

$$ a \cdot c = -70 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -70 $ and add to $ b = 6 $.

Step 5: All pairs of numbers with a product of $ -70 $ are:

PRODUCT = -70
-1     70	1     -70
-2     35	2     -35
-5     14	5     -14
-7     10	7     -10

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

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ 6 }$, we conclude the polynomial cannot be factored.

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